Find a matrix A such that 2 A - 3 B + 5 C = 0 , where B = - 2 2 0 3 1 4 and C = 2 0 - 2 7 1 6

We have, 2 A - 3 B + 5 C = O ⇒ 2 A = 3 B - 5 C ⇒ 2 A = 3 - 2 2 0 3 1 4 - 5 2 0 - 2 7 1 6 2 A = - 6 6 0 9 3 12 - 10 0 - 10 35 5 30 2 A = - 16 6 10 - 26 - 2 - 18 ⇒ A = - 8 3 5 - 13 - 1 - 9

Let P 1 = I = 1 0 0 0 1 0 0 0 1 , P 2 = 1 0 0 0 0 1 0 1 0 , P 3 = 0 1 0 1 0 0 0 0 1 , P 4 = 0 1 0 0 0 1 1 0 0 , P 5 = 0 0 1 1 0 0 0 1 0 , P 6 = 0 0 1 0 1 0 1 0 0 , and X = ∑ k = 1 6 P K 2 1 3 1 0 2 3 2 1 P K T where P K T denotes the transpose of the matrix P K . Then which of the following options is/are correct?

X - 30 I is an invertible matrix

If A = k l m n and k n ≠ l m , then the value of A 2 - ( k + n ) A + ( k n - l m ) I equals

The zero matrix of order 2 × 2

2 I where I is the identity matrix of order 2 × 2 .

Using the properties of determinants show that x p q p x q q q x = x - p x 2 + p x - 2 q 2 .

Consider, LHS = x p q p x q q q x Applying C 1 → C 1 - C 2 = x - p p q p - x x q 0 q x = x - p p q - x - p x q 0 q x = x - p 1 p q - 1 x q 0 q x [Taking x - p common from C 1 ] Applying R 2 → R 2 + R 1 = x - p 1 p q 0 x + p 2 q 0 q x = x - p x + p x - 2 q 2 [Expanding along C 1 ] = x - p x 2 + p x - 2 q 2 = RHS Hence proved.

EASY

Earn 100

Give an example to show that the product of two non-zero matrices is a zero matrix.

Important Questions on Matrices

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

P = [\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}]

a b c = 1

P^{T} P = I

then the value of

a^{3} + b^{3} + c^{3}

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

A

is a square matrix such that

A^{2} = I

, then

(A - I)^{3} + (A + I)^{3} - 7 A

is equal to

HARD

Mathematics>Algebra>Matrices>Algebra of Matrices

Let $a, b, c \in R$ be all non-zero and satisfies $a^{3} + b^{3} + c^{3} = 2$ . If the matrix $A = [\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}] \begin{array}{l} \end{array}$ satisfies $A^{T} A = I,$ then a value of $a b c$ can be

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

Let

A

and

B

be two

3 \times 3

real matrices such that

(A^{2} - B^{2})

is invertible matrix. If

A^{5} = B^{5}

and

A^{3} B^{2} = A^{2} B^{3},

then the value of the determinant of the matrix

A^{3} + B^{3}

is equal to :

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

A = [\begin{matrix} 5 a & - b \\ 3 & 2 \end{matrix}]

and

A . a d j A = A A^{T}

, then

5 a + b

is equal to

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

Let

P

and

Q

3 \times 3

matrices with

P \neq Q .

P^{3} = Q^{3}

and

P^{2} Q = Q^{2} P,

then determinant of

(P^{2} + Q^{2})

is equal to

EASY

Mathematics>Algebra>Matrices>Algebra of Matrices

3 A + 4 B^{'} = [\begin{matrix} 7 & - 10 & 17 \\ 0 & 6 & 31 \end{matrix}]

and

2 B - 3 A^{'} = [\begin{matrix} - 1 & 18 \\ 4 & 0 \\ - 5 & - 7 \end{matrix}]

then

B =

EASY

Mathematics>Algebra>Matrices>Algebra of Matrices

Find a matrix

A

such that

2 A - 3 B + 5 C = 0

, where

B = [\begin{array}{r} - 2 & 2 & 0 \\ 3 & 1 & 4 \end{array}]

and

C = [\begin{array}{r} 2 & 0 & - 2 \\ 7 & 1 & 6 \end{array}]

EASY

Mathematics>Algebra>Matrices>Algebra of Matrices

The number of all

3 \times 3

matrices

A,

with entries from the set

\{- 1, 0, 1\}

such that the sum of the diagonal elements of

A A^{T}

3,

is ___________.

HARD

Mathematics>Algebra>Matrices>Algebra of Matrices

Let

P_{1} = I = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], P_{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}], P_{3} = [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}],

P_{4} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}], P_{5} = [\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}], P_{6} = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}],

and

X = \sum_{k = 1}^{6} P_{K} [\begin{matrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{matrix}] P_{K}^{T}

where

P_{K}^{T}

denotes the transpose of the matrix

P_{K} .

Then which of the following options is/are correct?

HARD

Mathematics>Algebra>Matrices>Algebra of Matrices

Let

A = [a_{i j}]

be a real matrix of order

3 \times 3,

such that

a_{i 1} + a_{i 2} + a_{i 3} = 1,

for

i = 1, 2, 3 .

Then, the sum of all the entries of the matrix

A^{3}

is equal to:

HARD

Mathematics>Algebra>Matrices>Algebra of Matrices

A

is a

3 \times 3

non-singular matrix such that

A A^{'} = A^{'} A

and

B = A^{- 1} A^{'},

then

B B^{'}

equals, where

X^{'}

denotes the transpose of the matrix

X

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

Let

A, B, C, D

be square real matrices such that

C^{T} = D A B, D^{T} = A B C, S = A B C D

, then

S^{2}

is equal to

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

A = [\begin{array}{l} k & l \\ m & n \end{array}]

and

k n \neq l m,

then the value of

A^{2} - (k + n) A + (k n - l m) I

equals

EASY

Mathematics>Algebra>Matrices>Algebra of Matrices

Let $A$ be a $4 \times 4$ matrix with real entries. Consider the sets

$K = \{(\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}) : A (\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix})\},$

$J = \{(\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \end{matrix}) : (\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \end{matrix}) = A (\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{matrix})\}$ for some $y_{1}, y_{2}, y_{3} . y_{4} \in R\}$

Suppose $K = J$ . Then, which one of the following statements is necessarily true?

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

Using the properties of determinants show that

$|\begin{matrix} x & p & q \\ p & x & q \\ q & q & x \end{matrix}| = (x - p) (x^{2} + p x - 2 q^{2})$ .

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

A

is a square matrix such that

A^{2} = A

, then

(I - A)^{3} + A

is equal to

HARD

Mathematics>Algebra>Matrices>Algebra of Matrices

The total number of matrices

A = [\begin{matrix} 0 & 2 y & 1 \\ 2 x & y & - 1 \\ 2 x & - y & 1 \end{matrix}], (x, y \in R, x \neq y)

for which

A^{T} A = 3 I_{3}

is:

HARD

Mathematics>Algebra>Matrices>Algebra of Matrices

For two

3 \times 3

matrices

A

and

B

, let

A + B = 2 B^{'}

and

3 A + 2 B = I_{3},

where

B^{'}

is the transpose of

B

and

I_{3}

3 \times 3

identity matrix. Then :

MEDIUM

Mathematics>Algebra>Matrices>Algebra of Matrices

If $A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{matrix}]$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

Give an example to show that the product of two non-zero matrices is a zero matrix.

Important Questions on Matrices

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