If A = - 1 1 2 3 0 4 2 1 - 2 and B = 2 - 3 - 4 - 1 0 - 1 0 - 1 0 , then examine whether the matrix A 2 - 2 B is singular.

Given that, the matrices are A = - 1 1 2 3 0 4 2 1 - 2 and B = 2 - 3 - 4 - 1 0 - 1 0 - 1 0 . Consider, A 2 = - 1 1 2 3 0 4 2 1 - 2 - 1 1 2 3 0 4 2 1 - 2 = 8 1 - 2 5 7 - 2 - 3 0 12 Now consider, A 2 - 2 B = 8 1 - 2 5 7 - 2 - 3 0 12 - 2 2 - 3 - 4 - 1 0 - 1 0 - 1 0 = 8 1 - 2 5 7 - 2 - 3 0 12 - 4 - 6 - 8 - 2 0 - 2 0 - 2 0 = 4 7 6 7 7 0 - 3 2 12 Now, consider the determinant of A 2 - 2 B A 2 - 2 B = 4 7 6 7 7 0 - 3 2 12 = - 42 Since, determinant is not zero the matrix A 2 - 2 B is not singular.

Construct a 2 × 2 matrix whose elements are given by, a i j = | 2 i - 3 j | 3

Let A be the required matrix. It is a 2 × 2 matrix. So it has 2 rows and 2 columns. Given a i j = | 2 i - 3 j | 3 . So, a 11 = ( 2 × 1 ) - 3 × 1 3 = 1 3 a 12 = ( 2 × 1 ) - 3 × 2 3 = 4 3 a 21 = ( 2 × 2 ) - 3 × 1 3 = 1 3 a 22 = ( 2 × 2 ) - 3 × 2 3 = 2 3 In general, a matrix of 2 × 2 order given by A = a 11 a 12 a 21 a 22 . So, the matrix is A = 1 3 4 3 1 3 2 3 .

If matrix A = a i j 3 × 2 and a i j = 3 i - 2 j 2 of matrix A , then find A .

Given that A = a i j 3 × 2 ⇒ A = a 11 a 12 a 21 a 22 a 31 a 32 3 × 2 And also given that a i j = 3 i - 2 j 2 Now, a 11 = 3 1 - 2 1 2 = 3 - 2 2 = 1 a 12 = 3 1 - 2 2 2 = 3 - 4 2 = 1 a 21 = 3 2 - 2 1 2 = 6 - 2 2 = 16 a 22 = 3 2 - 2 2 2 = 6 - 4 2 = 4 a 31 = 3 3 - 2 1 2 = 9 - 2 2 = 49 a 32 = 3 3 - 2 2 2 = 9 - 4 2 = 25 Therefore, A = 1 1 16 4 49 25 3 × 2 .

Construct a 3 × 2 matrix whose elements are given by a ij = 1 2 [ i − 3 j ] .

In general 3 × 2 matrix is given by A = a 11 a 12 a 21 a 22 a 31 a 32 Now a ij = 1 2 | i - 3 j | , i = 1 , 2 , 3 j = 1 , 2 Therfore a 11 = 1 2 | 1 - 3 × 1 | = 1 , a 12 = 1 2 ∣ 1 - 3 × 21 = 5 2 a 21 = 1 2 2 - 3 × 1 = 1 2 , a 22 = 1 2 ∣ 2 - 3 × 2 I = 2 a 31 = 1 2 | 3 - 3 × 1 | = 0 , a 32 = 1 2 | 3 - 3 × 2 | = 3 2 Hence the required matrix is given by A = 1 5 2 1 2 2 0 3 2

EASY

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A square matrix is such that $A^{2} = I$ , find the value of ${(A - I)}^{3} + {(A + I)}^{3} - 7 A$ in simplified form.

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Important Questions on Matrices

EASY

Mathematics>Algebra>Matrices>Types of Matrices

Given

[\begin{matrix} 4 & 2 \\ - 1 & 1 \end{matrix}] M = 6 I

, where

M

is a matrix and

I

is unit matrix of order

2 \times 2

. If the order of matrix

M

p \times q

, then write the value of

p + q

HARD

Mathematics>Algebra>Matrices>Types of Matrices

Let

A = (a_{i j})

be an

n \times n

matrix defined by

a_{i j} = \{\begin{array}{lc} k^{i}, \forall i = j \\ 0, otherwise \end{array}

. If

m =

trace of

A

and

\lim_{k \to 1} \frac{n - m}{1 - k} = 171

then the value of

n

EASY

Mathematics>Algebra>Matrices>Types of Matrices

M = [m_{i j}]

a

4 \times 4

square matrix, where

i = 1, 2, 3, 4

and

j = 1, 2, 3, 4 .

If in the matrix

M, m_{i j} = \sin (i \times j) x

, then the value of the determinant

| M |

x = \frac{π}{3}

is equal to

HARD

Mathematics>Algebra>Matrices>Types of Matrices

The number of

3 \times 3

matrices

M

with entries from

{0,1, 2}

, such that the sum of the diagonal elements of

M^{T} M

5

, are

EASY

Mathematics>Algebra>Matrices>Types of Matrices

If $A = [\begin{array}{r} - 1 & 1 & 2 \\ 3 & 0 & 4 \\ 2 & 1 & - 2 \end{array}]$ and $B = [\begin{array}{r} 2 & - 3 & - 4 \\ - 1 & 0 & - 1 \\ 0 & - 1 & 0 \end{array}]$ , then examine whether the matrix $A^{2} - 2 B$ is singular.

EASY

Mathematics>Algebra>Matrices>Types of Matrices

For a

3 \times 4

matrix, elements are given by

a_{i j} = | - 3 i + 4 j |

, then

\sum_{i = 1}^{3} {(a_{i i})}^{i} =___

MEDIUM

Mathematics>Algebra>Matrices>Types of Matrices

The number of possible matrices of order

3 \times 3

with each entry

0

1

EASY

Mathematics>Algebra>Matrices>Types of Matrices

Construct a

2 \times 2

matrix whose elements are given by,

a_{i j} = \frac{| 2 i - 3 j |}{3}

EASY

Mathematics>Algebra>Matrices>Types of Matrices

A = {[a_{i j}]}_{n \times n}

such that

a_{i j} = 0

, for

i \neq j

, then

A

is _____

(a_{i i} \neq a_{j j}), (n > 1)

MEDIUM

Mathematics>Algebra>Matrices>Types of Matrices

A square matrix in which all the non-diagonal entries are zero and diagonal entries are same numbers is called _____ matrix

EASY

Mathematics>Algebra>Matrices>Types of Matrices

If matrix

A = {[a_{i j}]}_{3 \times 2}

and

[a_{i j}] = {(3 i - 2 j)}^{2}

of matrix

A

, then find

A

MEDIUM

Mathematics>Algebra>Matrices>Types of Matrices

Let

A + 2 B = [\begin{matrix} 1 & 2 & 0 \\ 6 & - 3 & 3 \\ - 5 & 3 & 1 \end{matrix}]

and

2 A - B = [\begin{matrix} 2 & - 1 & 5 \\ 2 & - 1 & 6 \\ 0 & 1 & 2 \end{matrix}] .

T r (A)

denotes the sum of all diagonal elements of the matrix

A,

then

Tr (A) - Tr (B)

has value equal to

EASY

Mathematics>Algebra>Matrices>Types of Matrices

Construct a

3 \times 2

matrix whose elements are given by

a_{ij} = \frac{1}{2} [i - 3 j]

MEDIUM

Mathematics>Algebra>Matrices>Types of Matrices

Given

[\begin{matrix} 4 & 2 \\ - 1 & 1 \end{matrix}] M = 6 I

, where

M

is the matrix and

I

is the unit matrix of order

2 \times 2

. Find the matrix

M

EASY

Mathematics>Algebra>Matrices>Types of Matrices

If matrix

A

{(a_{i j})}_{m \times n}

and

B

{(b_{j k})}_{n \times p}

then the order of the matrix

AB

EASY

Mathematics>Algebra>Matrices>Types of Matrices

A = [\begin{matrix} \begin{matrix} 1 & 0 & 2 \\ 5 & 1 & x \\ 1 & 1 & 1 \end{matrix} \end{matrix}]

is a singular matrix, then the value of

x

EASY

Mathematics>Algebra>Matrices>Types of Matrices

Let

A = [a_{i j}]

be a

2 \times 2

matrix with

a_{i j} = (- 1)^{j} i^{2} .

Then the matrix

A

is given by

EASY

Mathematics>Algebra>Matrices>Types of Matrices

Consider the following statements :
(I) There exists a square matrix with

12

elements
(II) There exists a zero matrix with

12

elements

MEDIUM

Mathematics>Algebra>Matrices>Types of Matrices

A = [\begin{matrix} a^{2} & 15 & 31 \\ 12 & b^{2} & 41 \\ 35 & 61 & c^{2} \end{matrix}]

and

B = [\begin{matrix} 2 a & 3 & 5 \\ 2 & 2 b & 8 \\ 1 & 4 & 2 c - 3 \end{matrix}]

are two matrices such that the sum of the principal diagonal elements of both

A

and

B

are equal, then the product of the principal diagonal elements of

B

EASY

Mathematics>Algebra>Matrices>Types of Matrices

The least positive integer

n

such that

{(\begin{matrix} \cos \frac{π}{4} & \sin \frac{π}{4} \\ - \sin \frac{π}{4} & \cos \frac{π}{4} \end{matrix})}^{n}

is an identity matrix of order

2

A square matrix is such that A2=I, find the value of A-I3+A+I3-7A in simplified form.

Important Questions on Matrices

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A square matrix is such that $A^{2} = I$ , find the value of ${(A - I)}^{3} + {(A + I)}^{3} - 7 A$ in simplified form.