Find the matrix $X$ so that $X\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right]$

1. Introduction to Matrices :

A set of numbers (real or imaginary) or symbols or expressions arranged in the form of a rectangular array of $m$ rows and $n$ columns is called $m\times n$ matrix.

2. Types of Matrices :

(i) A matrix having only one row is called a row matrix.

(ii) A matrix having only one column is called a column matrix.

(iii) A matrix in which the number of rows is equal to the number of columns, say $n$, is called a square matrix of order $n$.

(iv) The elements $a_{ij}$ of a square matrix $A={\left[a_{ij}\right]}_{n\times n}$ for which $i=j$ i.e. the elements $a_{11},a_{22},....,a_{nn}$ are called the diagonal elements and the line along which they lie is called the principal diagonal or leading diagonal.

(v) A square matrix $A={\left[a_{ij}\right]}_{n\times n}$ is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero i.e. $a_{ij}=0$ for $i\ne j.$

(vi) A square matrix $A={\left[a_{ij}\right]}_{n\times n}$ is called a scalar matrix, if $a_{ij}=0$ for all $i\ne j$ and $a_{ij}=c$ for all $i=j,$ where $c\ne 0.$

(vii) A square matrix $A={\left[a_{ij}\right]}_{n\times n}$ is called an identity or a unit matrix, if $a_{ij}=0$ for all $i\ne j$ and $a_{ij}=1$ for all $i=j.$

(viii) A matrix whose all elements are zero is called a null matrix or a zero matrix.

(ix) A square matrix $A=\left[a_{ij}\right]$ is called

(a) An upper triangular matrix, if $a_{ij}=0$ for all $i>j$

(b) A lower triangular matrix, if $a_{ij}=0$ for all $i<j$

(x) Equality of matrices:  Two matrices $A={\left[a_{ij}\right]}_{m\times n}$ and $B={\left[b_{ij}\right]}_{m\times n}$ of the same order are equal, if $a_{ij}=b_{ij}$ for all $i=1,2, \dots ,m; j=1,2, \dots ,n$

3. Matrix addition and scalar multiplication

(i) If $A={\left[a_{ij}\right]}_{m\times n}$ and $B={\left[b_{ij}\right]}_{m\times n}$ are two matrices of the same order $m\times n,$ then their sum $A+B$ is an $m\times n$ matrix such that $\left(A+B\right)={\left[a_{ij}+b_{ij}\right]}_{m\times n}$ for $i=1,2,...,m$ and $j=1,2,3,...,n$.

(ii) Properties of matrix addition

(a) Commutativity: If $A$ and $B$ are two matrices of the same order, then $A+B=B+A.$

(b) Associativity: If $A,B,C$ are three matrices of the same order, then $\left(A+B\right)+C=A+\left(B+C\right).$

(c) Existence of Identity: The null matrix is the identity element for matrix addition i.e., $A+O=A=O+A$

(d) Existence of Inverse: For every matrix $A={\left[a_{ij}\right]}_{m\times n},$ there exists a matrix $-A={\left[-a_{ij}\right]}_{m\times n},$ such that $A+\left(-A\right)=O=\left(-A\right)+A.$

(e) Cancellation Laws: If $A,B,C$ are three matrices of the same order, then $A+B=A+C\Rightarrow B=C$ and, $B+A=C+A\Rightarrow B=C.$

(iii) Scalar multiplication of matrices
Let $A=\left[a_{ij}\right]$ be an $m\times n$ matrix and $k$ be any scalar number. Then, the matrix obtained by multiplying every element of $A$ by $k$ is called the scalar multiple of $A$ by $k$ and is denoted by $kA.$ Thus, $kA={\left[k{a}_{ij}\right]}_{m\times n}$

(iv) Properties of scalar multiplication of matrices: If $A,B$ are two matrices of the same order and $k,l$ are scalars, then

(a) $k\left(A+B\right)=kA+kB$

(b) $\left(k+l\right)A=kA+lA$

(c) $\left(kl\right)A=k\left(lA\right)=l\left(kA\right)$

(d) $\left(-k\right)A=-\left(kA\right)=k\left(-A\right)$

(iv) If $A$ and $B$ are two matrices of the same order, then $A-B=A+\left(-B\right).$

4. Matrix multiplication and its properties

(i) Two matrices $A$ and $B$ are conformable for the product $AB$ if the number of columns in $A$ is same as the number of rows in $B$. If $A={\left[a_{ij}\right]}_{m\times n}$ and $B={\left[b_{ij}\right]}_{n\times p}$ are two matrices, then $AB$ is a $m\times p$ matrix such that ABij=∑r=1nairbrj

(ii) Matrix multiplication is not commutative.

(iii) Matrix multiplication is associative i.e., $\left(AB\right)C=A\left(BC\right)$ wherever both sides of the equality are defined.

(iv) Matrix multiplication is distributive over matrix addition

i.e. $A(B+C)=AB+AC$ and $(B+C)A=BA+CA$ wherever both sides of the equality are defined.

(v) If $A$ is $n\times n$ matrix, then $l_{n}A=A=A{l}_n.$

(vi) If $A$ is $m\times n$ matrix and $O$ is a null matrix, then $A_{m\times n}\times O_{n\times p}=O_{m\times p}$ and $O_{p\times m}\times A_{m\times n}=O_{p\times n}$ 

(vii) If $A$ is a square matrix, then we define $A^{n+1}=A^{n}A$

5. Transpose of a matrix

(i) Let $A=\left[a_{ij}\right]$ be an $m \times n$ matrix. Then, the transpose of $A,$ denoted by $A^T,$ is an $n \times m$ matrix such that $A^T=\left[a_{ji}\right]$ for all $i=1,2,\dots m, ; j=1,2,...,n$

(ii) Following are the properties of transpose of a matrix:

(a) ${\left(A^T\right)}^T=A$

(b) ${\left(A+B\right)}^T=A^T+B^T$

(c) ${\left(KA\right)}^T=K{A}^T$(where $K$ is any number or symbol)

(d) ${\left(AB\right)}^T=B^T{A}^T$

(e) ${\left(ABC\right)}^T=C^T{B}^T{A}^T$

6. Symmetric and Skew symmetric matrices- Definition and properties

(i) A square matrix $A=\left[a_{ij}\right]$ is called a symmetric matrix, if $a_{ij}=a_{ji}$ for all $i,j$ i.e., $A=A^T.$

(ii) A square matrix $A=\left[a_{ij}\right]$ is called a skew symmetric matrix, if $a_{ij}=-a_{ji}$ for all $i,j$ i.e., $A^T=-A.$

(iii) All main diagonal elements of a skew–symmetric matrix are zero.

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

(v) All positive integral powers of a symmetric matrix are symmetric matrices.

(vi) All odd positive integral powers of a skew–symmetric matrix are skew–symmetric matrices.

7. Elementary operations (transformation) of a matrix.

(i) Interchange of any two rows (columns).

(ii) Multiplying all elements of a row (column) of a matrix by a non-zero scalar.

(iii) Adding to the elements of a row (column), the corresponding elements of any other row (column) multiplied by any scalar.

(iv) A matrix obtained from an identity matrix by a single elementary operation is called an elementary matrix.

(v) Every elementary row (column) operation on an $m \times n$ matrix (not identity matrix) can be obtained by pre-multiplication (post-multiplication) with the corresponding elementary matrix obtained from the identity matrix $I_m\left(I_n\right)$ by subjecting it to the same elementary row (column) operation.

8. Inverse of a matrix by elementary operations   

(i) In order to find the inverse of a non-singular square matrix $A$ by elementary operations, we write $A=IA$ or $A=AI$

(ii) We perform a sequence of elementary row operations successively on $A$ on the LHS and the pre-factor $I$ on RHS till we obtain. The matrix $B,$ so obtained, is the desired inverse of matrix $A.$