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1

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.

2

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.

3

WB JEE 2008

MCQ (Single Correct Answer)

If the matrix $$\left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ is commutative with the matrix $$\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]$$ then

A
a = 0, b = c
B
b = 0, c = d
C
c = 0, d = a
D
d = 0, a = b

Explanation

Matrix $$\left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ is commutative with the matrix $$\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]$$

If we consider two matrices A and B are commutative on product then AB = BA

$$ \Rightarrow \left[ {\matrix{ a & b \cr c & d \cr } } \right]\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right] = \left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ a & b \cr c & d \cr } } \right]$$

$$ \Rightarrow \left[ {\matrix{ a & {a + b} \cr c & {c + d} \cr } } \right] = \left[ {\matrix{ {a + c} & {b + d} \cr c & d \cr } } \right]$$

Comparing corresponding elements of two matrices, we get

$$\Rightarrow$$ a + c = a, a + b = b + d and c + d = d

$$\Rightarrow$$ c = 0, a = d and c = 0.

4

WB JEE 2008

MCQ (Single Correct Answer)

The values of x for which the given matrix $$\left[ {\matrix{ { - x} & x & 2 \cr 2 & x & { - x} \cr x & { - 2} & { - x} \cr } } \right]$$ will be non-singular are

A
$$ - 2 \le x \le 2$$
B
for all x other than 2 and $$-$$2
C
$$x \ge 2$$
D
$$x \le - 2$$

Explanation

If matrix A is non singular then $$\left| A \right| \ne 0$$

$$\therefore$$ $$A = \left[ {\matrix{ { - x} & x & 2 \cr 2 & x & { - x} \cr x & { - 2} & { - x} \cr } } \right]$$

$$\left| A \right| = \left[ {\matrix{ { - x} & x & 2 \cr 2 & x & { - x} \cr x & { - 2} & { - x} \cr } } \right]$$

$$ = - x( - {x^2} - 2x) - x( - 2x + {x^2}) + 2( - 4 - {x^2}) = 2{x^2} - 8$$

$$\therefore$$ $$\left| A \right| \ne 0$$

$$ \Rightarrow 2({x^2} - 4) \ne 0 \Rightarrow x \ne 2,\, - 2$$

$$\therefore$$ given matrix will be non singular for all x other than 2 and $$-$$2.

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