1. Basics of Matrices:

Definition: A rectangular array of $m n$ numbers is known as matrix. Unlike determinants, it has no value.

It is represented as

$A = [\begin{matrix} \begin{matrix} a_{11} & a_{12} \dots & a_{1 n} \\ a_{21} & a_{22} \dots & a_{2 n} \\ : & : & : \end{matrix} \\ \begin{matrix} a_{m 1} & a_{m 2} \dots & a_{m n} \end{matrix} \end{matrix}]$

which is also abbreviated as: $A = [a_{i j}], 1 \leq i \leq m; 1 \leq j \leq n$ .

Here, $a_{i j}$ denotes the element of $i$ th row and $j$ th column. $m \times n$ is known as order of matrix $A$ . $a_{11}$ , $a_{12}$ , $a_{13}$ , … are known as elements of first row $R_{1}$ . Similarly, $a_{11}$ , $a_{21}$ , $a_{31}$ , … are known as elements of first column $C_{1}$ and so on.

Diagonal Elements:

The elements $a_{11}, a_{22}, a_{33}, \dots$ are called diagonal elements. The line along which the diagonal elements lie is called diagonal or main diagonal or principal diagonal of the matrix.

Trace of a Matrix:

Sum of diagonal elements of a matrix is known as trace of the matrix. Trace of a matrix $A$ is denoted as $t r (A)$ .

Properties of Trace:

(i) $t r (A + B) = t r (A) + t r (B)$

(ii) $t r (k A) = k . t r (A)$ , where $k$ is any scalar

(iii) $t r (A B) = t r (B A)$

(iv) $t r (A) = t r (A^{T})$ , where $A^{T}$ denotes transpose of $A$

Types of Matrices

(i) Row Matrix: A matrix having one row is known as row matrix. It is also known as row vector.

e.g.: $A = [a_{11}, a_{12, \dots . .} a_{1 n}]$

(ii) Column Matrix: A matrix having one row is known as column matrix. It is also known as column vector.

e.g.: $A = [\begin{matrix} \begin{matrix} a_{11} \\ a_{21} \\ : \end{matrix} \\ a_{m 1} \end{matrix}]$

(iii) Zero or Null Matrix:

Any $m \times n$ matrix whose all entries are zero is known as zero or null matrix. It is denoted as $O_{m \times n}$ or $O$ .

e.g.: $O = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}]$ is a $3 \times 2$ null matrix.

(iv) Horizontal Matrix: A matrix of order $m \times n$ is a horizontal matrix if $n > m .$

e.g.: $[\begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 1 \end{matrix}]$ .

(v) Vertical Matrix: A matrix of order $m \times n$ is a vertical matrix if $m > n .$

e.g.: $[\begin{matrix} \begin{matrix} 2 & 5 \\ 1 & 1 \\ 3 & 6 \end{matrix} \\ \begin{matrix} 2 & 4 \end{matrix} \end{matrix}]$ .

(vi) Square Matrix:

If number of rows in a matrix is same as number of columns in the matrix, then the matrix is known as square matrix. If a square matrix has $n$ rows and $n$ columns, then its order is generally written as $n$ (not $n \times n$ ).

In a square matrix the pair of elements $a_{i j} & a_{j i}$ are called conjugate elements.

(vii) Diagonal Matrix:

A diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero.

e.g., $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix}]$ which is also represented as $diag (1,2, 1)$

(viii) Scalar Matrix:

A diagonal matrix becomes scalar matrix if all the diagonal elements are equal.

e.g., $A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}]$

(ix) Unit Matrix or Identity Matrix:

A diagonal matrix becomes unit matrix if all the diagonal elements are equal to unity.

e.g., $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ . It is denoted as $I or I_{3} or I_{3 \times 3}$ .

(x) Upper and Lower Triangular Matrix:

A square matrix is called lower triangular matrix if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular matrix if all the entries below the main diagonal are zero.

e.g., $A = [\begin{matrix} 1 & 2 & 3 \\ 0 & 4 & 0 \\ 0 & 0 & - 1 \end{matrix}]$ , $B = [\begin{matrix} - 1 & 0 & 0 \\ 5 & 2 & 0 \\ 3 & 1 & 2 \end{matrix}]$ . Here $A$ is upper triangular matrix and $B$ is lower triangular matrix.

2. Algebra of Matrices

(i) Equality of Matrices:

Let $A = [a_{i j}] & B = [b_{i j}]$ be two matrices. Then they are said to be equal if both have the same order and $a_{i j} = b_{i j}$ for each pair of $i & j .$

(ii) Addition and Subtraction:

$A \pm B = [a_{i j}] \pm [b_{i j}] = [a_{i j} \pm b_{i j}]$ , where $A & B$ are of the same order matrices.

Properties of Matrices Addition:

(a) Matrix addition is commutative:

$A + B = B + A,$

(b) Matrix addition is associative:

$(A + B) + C = A + (B + C)$

(iii) Multiplication of a Matrix by a Scalar:

Let $A = [a_{i j}]$ . Then $k A = k [a_{i j}] = [k a_{i j}]$ . For example, if $A = [\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ , then $k A = [\begin{matrix} k a & k b & k c \\ k b & k c & k a \\ k c & k a & k b \end{matrix}]$ .

(iv) Multiplication of Matrices:

If $A = {[a_{i j}]}_{m \times p} & B = {[b_{i j}]}_{p \times n}$ matrix, then $A B$ is defined as $A B = {[Σ_{r = 1}^{p} a_{i r} . b_{r j}]}_{m \times n}$ for all pairs of $i$ and $j$ .

This is generally known as row by column multiplication method.

Note: Product $A B$ is defined if and only if number of columns in $A$ is same as number of rows in $B$ .

For example, if matrix $A$ has order $3 \times 4$ and $B$ has order $4 \times 5$ , then the product is defined but if $A$ has order $3 \times 4$ and $B$ has order $3 \times 4$ , then product $A B$ is not defined.

Properties of Matrix Multiplication:

(a) In general matrix multiplication is not commutative.

$\Rightarrow A B \neq B A$ .

But sometimes they may be commute. Like $A I = I A = A$ , $A O = O A = O$ and $A^{- 1} A = I = A A^{- 1}$

(b) Matrix Multiplication is Associative:

If $A, B & C$ are conformable for the product $A B & B C,$ then

$(A \cdot B) \cdot C = A \cdot (B \cdot C)$

$A (B + C) = A B + A C$ and $(A + B) C = A C + B C$ , provided $A, B & C$ are conformable for respective products.

(d) Positive Integral Powers of a Square Matrix:

For a square matrix $A,$ powers are defined as usual.

Like $A^{2} A = (A A) A = A (A A) = A^{3} .$

Note that for a unit matrix $I$ of any order, $I^{m} = I$ for all $m \in N .$

(v) Matrix Polynomial:

If $f (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots a_{n - 1} x^{n - 1} + a_{n} x^{n}$ , then we define a matrix polynomial as

$f (A) = a_{0} I + a_{1} A + a_{2} A^{2} + \dots a_{n - 1} A^{n - 1} + a_{n} A^{n}$ where $A$ is the given square matrix.

(vi) Transpose of A Matrix:

Let $A = [a_{i j}]$ be any matrix of order $m \times n$ , then $A^{T} o r A^{'} = [a_{j i}]$ with order $n \times m$ . For $1 \leq i \leq n & 1 \leq j \leq m$ is known as transpose of matrix $A$ .

Properties of Transpose:

If $A^{T} & B^{T}$ denote the transpose of $A$ and $B$ , then we have,

(a) ${(A \pm B)}^{T} = A^{T} \pm B^{T}$ ; $A & B$ must be of the same order.

(b) ${(A B)}^{T} = B^{T} A^{T}; A & B$ are conformable for matrix product $A B .$

In general, ${(A_{1} \cdot A_{2} \cdot \dots \cdot A_{n})}^{T} = A_{n}^{T} \cdot \dots \cdot A_{2}^{T} \cdot A_{1}^{T}$

(d) ${(k A)}^{T} = k A^{T}$ where $k$ is any scalar.

3. Symmetric & Skew Symmetric Matrix:

A square matrix is said to be symmetric if $A^{T} = A$ and it is skew symmetric if $A^{T} = - A$ .

In other words, for symmetric matrix $A = [a_{i j}]$ , we have $a_{i j} = a_{j i}$ , for all $i$ and $j$ .

And for skew symmetric matrix $A = [a_{i j}]$ , we have $a_{i j} = - a_{j i}$ , for all $i$ and $j$ . For diagonal elements, we have $i = j$ which implies $a_{i i} = - a_{i i} \Rightarrow a_{i i} = 0$ . Thus, all the diagonal elements of a skew symmetric matrix are always zero.

Properties of Symmetric & Skew Symmetric Matrix:

(i) $A + A^{T}$ is symmetric matrix and $A - A^{T}$ is skew symmetric matrix for any square matrix $A$ .

(ii) The sum of two symmetric matrix is symmetric matrix and the sum of two skew symmetric matrix is skew symmetric matrix.

Let $A$ and $B$ be two symmetric matrices i.e., $A^{T} = A & B^{T} = B$ .

${(A + B)}^{T} = {(A)}^{T} + {(B)}^{T} = A + B$ . Similarly, we can prove for skew symmetric matrix.

(iii) Let $A$ and $B$ be two symmetric matrices i.e., $A^{T} = A, B^{T} = B$ , then

(a) $A B + B A$ is a symmetric matrix.

(b) $A B - B A$ is a skew symmetric matrix.

(iv) Every square matrix can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix.

For example, let $A$ be any square matrix. Then $A = \frac{1}{2} (A + A^{T}) + \frac{1}{2} (A - A^{T})$ , where $\frac{1}{2} (A + A^{T})$ is symmetric and $\frac{1}{2} (A - A^{T})$ is skew symmetric.

(v) If $A$ is any symmetric skew matrix, then $A^{3}, A^{5}, \dots$ are skew symmetric matrices while $A^{2}, A^{4}, A^{6}, \dots$ are symmetric matrices. But if $A$ is symmetric matrix, then $A, A^{2}, A^{3}, A^{4}, \dots$ are all symmetric matrices.

4. Determinants:

Every square matrix can be associated to an expression which is known as its determinant. If $A = [a_{i j}]$ is a square matrix of order $n,$ then the determinant of $A$ is denoted by det $A$ or $|A|$ or

$|\begin{matrix} \begin{matrix} a_{11} & a_{12} & \dots \\ a_{21} & a_{22} & \dots \\ ⋮ & ⋮ \end{matrix} \begin{matrix} a_{1 j} & \dots & a_{1 n} \\ a_{2 j} & \dots & a_{2 n} \end{matrix} \\ \begin{matrix} a_{i 1} & a_{i 2} & \dots \\ ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots \end{matrix} \begin{matrix} a_{i j} & \dots & a_{i n} \\ ⋮ & ⋮ \\ a_{n j} & \dots & a_{n n} \end{matrix} \end{matrix}|$

Only square matrices have determinants. The matrices which are not square do not have determinants.

Minors and cofactors:

(i) Minors:

Let $A = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]$ be any square matrix. Then Minor of element $a_{i j}$ of $A$ or $d e t (A)$ is denoted by $M_{i j}$ and is obtained by a determinant made by leaving $i^{t h}$ row and $j^{t h}$ column.

e.g., $M_{11} = |\begin{matrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix}|, M_{12} = |\begin{matrix} a_{21} & a_{23} \\ a_{31} & a_{31} \end{matrix}|$ .

(ii) Co-factors: Co-factor of element $a_{i j}$ of determinant $|A|$ is denoted by $A_{i j}$ or $C_{i j}$ and this is given by $A_{i j} = {(- 1)}^{i + j} M_{i j}$

Expansion of determinant:

Determinant can be expanded along any row or any column. The expansion of any determinant or value of determinant is the sum of product of elements and their co-factors of any row or any column.

Let $|A| = |\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}|$ .

Then expansion of $|A|$ along first row $|A| = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13}$ .

Expansion along first column $|A| = a_{11} A_{11} + a_{12} A_{21} + a_{31} A_{31} .$

5. Properties of Determinants:

(i) The value of determinant is not altered by changing the rows into columns and the columns into rows i.e.,

$|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}|$

Thus for any square matrix $A$ , we have $|A| = {|A|}^{T}$ .

(ii) If any two rows or columns of a determinants are (interchanged), the sign of the determinants gets changed, whereas the numerical value remains the same.

e.g., $|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = - |\begin{matrix} a_{2} & b_{2} & c_{2} \\ a_{1} & b_{1} & c_{1} \\ a_{3} & b_{3} & c_{3} \end{matrix}|$

(iii) If any two rows or two columns of a determinant are identical or their elements are proportional, the value of determinant is zero.

e.g., $|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{1} & b_{1} & c_{1} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = 0$

(iv) If any row or column of a determinant have all elements zero, the value of determinant is zero.

e.g., $|\begin{matrix} a_{1} & b_{1} & c_{1} \\ 0 & 0 & 0 \\ a_{3} & b_{3} & c_{3} \end{matrix}| = 0$

(v) If each element in any row or in any column is multiplied by a constant, then the value of the determinant is multiplied by the same constant

e.g., $|\begin{matrix} k a_{1} & k b_{1} & k c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = k |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|$ .

In other words, if a number is common in any row or column, then it can be treated as it is common to whole determinant. Similarly, if a scalar is multiplied in a determinant, then it is only multiplied in any one row or column.

Thus, we have, $|k A| = k^{n} |A|$ , where $k$ is any scalar and $n$ is order of $A$ or $|A|$ .

(vi) Determinant of odd order skew symmetric matrix is always zero.

(vii) $|A B| = |A| |B|$ .

In general,

$|A_{1} A_{2} A_{3} \dots A_{n}| = |A_{1}| |A_{2}| |A_{3}| \dots |A_{n}|$

(viii) If each element of any row or column consists of the sum of two terms, then the determinant can be expressed as the sum of two determinants of the same order with one set of terms in one determinant and other set of terms in other determinant and remaining elements are not changed.

e.g., $|\begin{matrix} a_{1} + α_{1} & b_{1} & c_{1} \\ a_{2} + α_{2} & b_{2} & c_{2} \\ a_{3} + α_{3} & b_{3} & c_{3} \end{matrix}| = |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| + |\begin{matrix} α_{1} & b_{1} & c_{1} \\ α_{2} & b_{2} & c_{2} \\ α_{3} & b_{3} & c_{3} \end{matrix}|$ .

(ix) If each element of a row or column of a determinant is added or subtracted by the same multiplier of the corresponding element of rows or columns, the determinant remains unaltered.

e.g., $|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = |\begin{matrix} a_{1} + h b_{1} + k c_{1} & b_{1} & c_{1} \\ a_{2} + h b_{2} + k c_{2} & b_{2} & c_{2} \\ a_{3} + h b_{3} + k c_{3} & b_{3} & c_{3} \end{matrix}|$ .

Here first column has changed but value of determinant has not changed. This operation is represented as $C_{1} \to C_{1} + h C_{2} + k C_{3}$ .

(x) Differentiation of a determinant:

It can be done either by row wise or column wise.

Let $∆ = |\begin{matrix} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ ϕ_{1} (x) & ϕ_{2} (x) & ϕ_{3} (x) \\ ψ_{1} (x) & ψ_{2} (x) & ψ_{3} (x) \end{matrix}|$

Then, $\frac{d ∆}{d x} = |\begin{matrix} {f_{1}}^{'} (x) & {f_{2}}^{'} (x) & {f_{3}}^{'} (x) \\ ϕ_{1} (x) & ϕ_{2} (x) & ϕ_{3} (x) \\ ψ_{1} (x) & ψ_{2} (x) & ψ_{3} (x) \end{matrix}| + |\begin{matrix} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ {ϕ_{1}}^{'} (x) & {ϕ_{2}}^{'} (x) & {ϕ_{3}}^{'} (x) \\ ψ_{1} (x) & ψ_{2} (x) & ψ_{3} (x) \end{matrix}| + |\begin{matrix} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ ϕ_{1} (x) & ϕ_{2} (x) & ϕ_{3} (x) \\ {ψ_{1}}^{'} (x) & {ψ_{2}}^{'} (x) & {ψ_{3}}^{'} (x) \end{matrix}|$

$\frac{d ∆}{d x} = |\begin{matrix} {f_{1}}^{'} (x) & f_{2} (x) & f_{3} (x) \\ {ϕ_{1}}^{'} (x) & ϕ_{2} (x) & ϕ_{3} (x) \\ {ψ_{1}}^{'} (x) & ψ_{2} (x) & ψ_{3} (x) \end{matrix}| + |\begin{matrix} f_{1} (x) & {f_{2}}^{'} (x) & f_{3} (x) \\ ϕ_{1} (x) & {ϕ_{2}}^{'} (x) & ϕ_{3} (x) \\ ψ_{1} (x) & {ψ_{2}}^{'} (x) & ψ_{3} (x) \end{matrix}| + |\begin{matrix} f_{1} (x) & f_{2} (x) & {f_{3}}^{'} (x) \\ ϕ_{1} (x) & ϕ_{2} (x) & {ϕ_{3}}^{'} (x) \\ ψ_{1} (x) & ψ_{2} (x) & {ψ_{3}}^{'} (x) \end{matrix}|$

(xi) Product of two determinants:

The product of two determinants $A$ and $B$ of same orders can be obtained by multiplying either the rows of $A$ by rows of $B$ or rows of $A$ by columns of B or columns of $A$ by column of $B$ or columns of $A$ by rows of $B$ . But for sake of convenience. To find the product $A B$ , the rows of $A$ are multiplied by the rows of B in the following manner:

If $A = |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|$ and $B = |\begin{matrix} α_{1} & β_{1} & γ_{1} \\ α_{2} & γ_{2} & γ_{2} \\ α_{3} & β_{3} & γ_{3} \end{matrix}|,$ then

$A B = |\begin{matrix} a_{1} α_{1} + b_{1} β_{1} + c_{1} γ_{1} & a_{1} α_{2} + b_{2} β_{2} + c_{1} γ_{2} & α_{1} α_{3} + b_{1} β_{3} + c_{1} γ_{3} \\ a_{2} α_{1} + b_{2} β_{1} + c_{2} γ_{1} & a_{2} α_{2} + b_{2} β_{2} + c_{2} γ_{2} & α_{2} α_{3} + b_{2} β_{3} + c_{2} γ_{3} \\ a_{3} α_{1} + b_{3} β_{1} + c_{3} γ_{1} & a_{3} α_{2} + b_{3} β_{2} + c_{3} γ_{2} & α_{3} α_{3} + b_{3} β_{3} + c_{3} γ_{3} \end{matrix}|$

Application of determinant:

Determinants are used in many areas of math and science. Some application includes finding area of triangle, checking collinearity of three points in coordinate geometry, etc. Determinants are also used to solve system of equations, which we will discuss later.

Singular and Non-singular Matrices:

A matrix $A$ is said to be singular or non-invertible matrix if $|A| = 0$ . If $|A| \neq 0$ , then it is known as non-singular or invertible matrix.

An important deduction:

If $A B = O$ , then at least one of them is null matrix or both are singular matrices.

6. Adjoint of A Square Matrix:

Let $A = {[a_{i j}]}_{3 \times 3} = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]$ be a square matrix, then a matrix obtained by replacing each element with its cofactor is known as cofactor matrix. It is represented by $A^{C}$ .

The transpose of $A^{C}$ is known as adjoint of $A$ . It is represented by $(a d j A)$ .

Thus, $(a d j A) = [\begin{matrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33} \end{matrix}]$ .

Properties of adjoint matrix:

If $A$ and $B$ are square matrices of order $n$ , then

(i) $A \cdot (a d j A) = |A| I = (a d j A) \cdot A$

(ii) $a d j (A B) = a d j B \cdot a d j A$

(iii) $|a d j A| = {|A|}^{n - 1}$

(iv) $a d j (a d j A) = {|A|}^{n - 2} A$ , where $|A| \neq 0$ .

(v) $|a d j (a d j (A))| = {|A|}^{{(n - 1)}^{2}}$

(vi) ${(a d j A)}^{T} = a d j A^{T}$

(vii) $a d j (A B) = (a d j B) \cdot (a d j A)$

(viii) $a d j (A^{n}) = {(a d j A)}^{n}, n \in N$

(ix) $a d j (k A) = k^{n - 1} (a d j A), k \in R$

(x) $a d j (O) = O$

(xi) $a d j (I_{n}) = I_{n}$

(xii) $A$ is symmetric matrix $\Rightarrow a d j A$ is a symmetric matrix.

(xiii) $A$ is diagonal matrix $\Rightarrow a d j A$ is a diagonal matrix.

(xiv) $A$ is triangular matrix $\Rightarrow a d j A$ is a triangular matrix.

7. Inverse of a Matrix:

A matrix $B$ is called the inverse of non-singular matrix $A$ , if $A B = I$ .

But we know, $A (a d j A) = |A| I$ , if $| A | \neq 0$ ,

Then we can write $\frac{1}{|A|} A (a d j A) = I \Rightarrow A (\frac{1}{|A|} a d j A) = I$ .

So, $A^{- 1} = \frac{1}{|A|} a d j A$ .

Properties of Inverse of Matrices:

(i) The inverse of a square matrix $A$ exists only if $| A | \neq 0$ .

(ii) ${(A^{- 1})}^{- 1} = A$

(iii) ${(A B)}^{- 1} = B^{- 1} A^{- 1}$

(iv) ${(k A)}^{- 1} = \frac{1}{k} A^{- 1}$ , where $k$ is any non-zero scalar.

(v) ${(A^{- 1})}^{T} = {(A^{T})}^{- 1}$

Special Types of Matrices

(i) Idempotent Matrix: A square matrix is idempotent provided if $A^{2} = A .$

(ii) Periodic Matrix: A square matrix is which satisfies the relation $A^{K - 1} = A,$ for some positive integer $K$ , is a periodic matrix. The period of the matrix is the least value of $K$ for which this holds true. Note that period of an idempotent matrix is $1$ .

(iii) Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order $m, m \in N, if A^{m} = O, A^{m - 1} \neq O .$

(iv) Involutory Matrix: The matrix is said to be an involuntary matrix if $A^{2} = I \Rightarrow A = A^{- 1}$

(v) Orthogonal Matrix: The matrix is said to be an orthogonal matrix if $A A^{T} = I Note that A^{T} = A^{- 1}$ for an orthogonal matrix.

8. System of Linear Equations:

(i) Matrix Method for solving System of Linear Equations:

$A X = B \Rightarrow A^{- 1} A X = A^{- 1} B$

$X = A^{- 1} B = \frac{(a d j A) . B}{|A|}$

Note:

(a) If $|A| \neq 0$ , then system is consistent having unique solution.

(b) If $|A| \neq 0$ & $(a d j A) \cdot B \neq O$ (Null matrix), then system is consistent having unique non-trivial solution.

(d) If $|A| = 0$ , matrix method fails.

(e) Now if $|A| = 0$ , $a d j A \cdot B = O$ , then system of equations will have infinite number of solutions provided corresponding ratios of variables are not equal.

(f) Now if $|A| = 0$ , $a d j A \cdot B = O$ , then system of equations will have infinite number of solutions

(ii) Determinant Method or Cramer's rule for solving System of Linear Equations:

Let the given the system of equation be

$a_{1} x + b_{1} y + c_{1} z = d_{1},$

$a_{2} x + b_{2} y + c_{2} z = d_{2}$ and

$a_{3} x + b_{3} y + c_{3} z = d_{3}$ .

Also, let $∆ = |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|$ ,

$∆_{x} = |\begin{matrix} d_{1} & b_{1} & c_{1} \\ d_{2} & b_{2} & c_{2} \\ d_{3} & b_{3} & c_{3} \end{matrix}|$ ,

$∆_{y} = |\begin{matrix} a_{1} & d_{1} & c_{1} \\ a_{2} & d_{2} & c_{2} \\ a_{3} & d_{3} & c_{3} \end{matrix}|$ and

$∆_{z} = |\begin{matrix} a_{1} & b_{1} & d_{1} \\ a_{2} & b_{2} & d_{2} \\ a_{3} & b_{3} & d_{3} \end{matrix}|$

(a) If $∆ = |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| \neq 0$ , system is consistent having unique solution and values of unknowns are $x = \frac{∆_{x}}{∆}, y = \frac{∆_{y}}{∆}, z = \frac{∆_{z}}{∆}$ .

(b) If $∆ = 0$ and at least one of $∆_{x}, ∆_{y}, ∆_{z}$ is non-zero, then given system of equation is inconsistent i.e., has no solution.

(c) If $∆ = 0$ and $∆_{x} = ∆_{y} = ∆_{z} = 0$ , then system of equations will have infinite number of solutions provided corresponding ratios of variables are not equal.

(d) If $∆ \neq 0$ and $∆_{x} = ∆_{y} = ∆_{z} = 0$ , then the given system of equations has unique solution $x = 0, y = 0$ and $z = 0,$ which is also called trivial solution.

If the area of the triangle with vertices 2,5, 7,k & 3,1 is 10, then find the value of k.

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

If the area of the triangle with vertices $(2, 5), (7, k) & (3, 1)$ is $10$ , then find the value of $k$ .