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1

### WB JEE 2010

MCQ (Single Correct Answer)

If the matrices $$A = \left[ {\matrix{ 2 & 1 & 3 \cr 4 & 1 & 0 \cr } } \right]$$ and $$B = \left[ {\matrix{ 1 & { - 1} \cr 0 & 2 \cr 5 & 0 \cr } } \right]$$, then AB will be

A
$$\left[ {\matrix{ {17} & 0 \cr 4 & { - 2} \cr } } \right]$$
B
$$\left[ {\matrix{ 4 & 0 \cr 0 & 4 \cr } } \right]$$
C
$$\left[ {\matrix{ {17} & 4 \cr 0 & { - 2} \cr } } \right]$$
D
$$\left[ {\matrix{ 0 & 0 \cr 0 & 0 \cr } } \right]$$

## Explanation

$$AB = \left( {\matrix{ 2 & 1 & 3 \cr 4 & 1 & 0 \cr } } \right)\left( {\matrix{ 1 & { - 1} \cr 0 & 2 \cr 5 & 0 \cr } } \right) = \left( {\matrix{ {17} & 0 \cr 4 & { - 2} \cr } } \right)$$

2

### WB JEE 2009

MCQ (Single Correct Answer)

If A and B are square matrices of the same order and AB = 3I, then A$$-$$1 is equal to

A
3B
B
$${1 \over 3}$$B
C
3B$$-$$1
D
$${1 \over 3}$$B$$-$$1

## Explanation

Given AB = 3I

$${A^{ - 1}}(AB) = {A^{ - 1}}(3I)$$ pre-multiplication by A$$-$$1

$$\Rightarrow {A^{ - 1}}AB = 3{A^{ - 1}}I$$

$$\Rightarrow IB = 3{A^{ - 1}}$$ ($$\because$$ $${A^{ - 1}}A = I$$)

$$\Rightarrow B = 3{A^{ - 1}} \Rightarrow {A^{ - 1}} = {1 \over 3}B$$

3

### WB JEE 2009

MCQ (Single Correct Answer)

If A2 $$-$$ A + I = 0, then the inverse of the matrix A is

A
A $$-$$ I
B
I $$-$$ A
C
A + I
D
A

## Explanation

A(A $$-$$ I) = $$-$$I

$$\Rightarrow$$ A(I $$-$$ A) = I $$\Rightarrow$$ A$$-$$1 = I $$-$$ A.

4

### WB JEE 2009

MCQ (Single Correct Answer)

If A is a square matrix. Then

A
A + AT is symmetric
B
AAT is skew-symmetric
C
AT + A is skew-symmetric
D
ATA is skew-symmetric

## Explanation

Let B = A + AT

$$\therefore$$ BT = (A + AT)T = AT + (AT)T = AT + A ($$\because$$ (A + B)T = BT + AT, (AT)T = A)

= A + AT = B (If AT = A, then A is symmetric)

$$\therefore$$ A + AT is symmetric.

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