Matrices: CBSE Class 12 Previous Year Questions

Which of the following properties is/are true for two matrices of suitable orders ?
(i) $(A+B{)}^{\prime }=A^{\prime }+B^{\prime }$
(ii) $(A-B{)}^{\prime }=B^{\prime }-A^{\prime }$
(iii) $(AB{)}^{\prime }=A^{\prime }{B}^{\prime }$
(iv) $(kAB{)}^{\prime }=k{B}^{\prime }{A}^{\prime }$ (k is a scalar)

1. (A)(i) only
2. (B)(i), (ii) and (iii)
3. (C)(i) and (ii)
4. (D)(i) and (iv)

If A and B are square matrices of same order, then which of the following statements is/are always true ?
(i) $(A+B)(A-B)=A^2-B^2$
(ii) $AB=BA$
(iii) $(A+B{)}^2=A^2+AB+BA+B^2$
(iv) $AB=0\Rightarrow A=0$ or $B=0$

1. (A)Only (i) and (iii)
2. (B)Only (ii) and (iii)
3. (C)Only (iii)
4. (D)Only (iii) and (iv)

If $A=\left[\begin{array}{ccc}1& a& b\\ -1& 2& c\\ 0& 5& 3\end{array}\right]$ is a symmetric matrix, then the value of $3a+b+c$ is

1. (A)2
2. (B)6
3. (C)4
4. (D)0

For a square matrix A, $(3A{)}^{-1}=$

1. (A)$3A^{-1}$
2. (B)$9A^{-1}$
3. (C)$\frac{1}{3}{A}^{-1}$
4. (D)$\frac{1}{9}{A}^{-1}$

If $A^2=4A+3I$ and $A^{-1}=xA+yI$, then the value of $(x+y)$ is :

1. (A)$-1$
2. (B)$1$
3. (C)$\frac{5}{3}$
4. (D)$7$

If A and B are skew-symmetric matrices of same order, then $A{B}^{\prime }+B{A}^{\prime }$ is a/an :

1. (A)symmetric matrix
2. (B)skew-symmetric matrix
3. (C)null matrix
4. (D)identity matrix

If a square matrix A is such that $A^2=A$ and $(I-A{)}^3=xA+I$, then value of x must be :

1. (A)$7$
2. (B)$5$
3. (C)$-7$
4. (D)$-1$

Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Let $A=\left[\begin{array}{ccc}0& -3& 4\\ 1& 0& 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}-3& 0& 1\\ 2& 4& 0\end{array}\right]$. If A + B + C = O, then matrix C is :

1. (A)$\left[\begin{array}{ccc}-3& -3& 5\\ 3& 4& 2\end{array}\right]$
2. (B)$\left[\begin{array}{ccc}3& 3& 5\\ -3& -4& -2\end{array}\right]$
3. (C)$\left[\begin{array}{ccc}3& 3& -5\\ -3& -4& -2\end{array}\right]$
4. (D)$\left[\begin{array}{ccc}-3& -3& -5\\ 3& 4& 2\end{array}\right]$

Given that $P=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right]$, $Q=\left[\begin{array}{cc}5& 2\\ 7& 4\end{array}\right]$ and $R=\left[\begin{array}{cc}2& 5\\ 3& 8\end{array}\right]$, find a matrix S such that $PQ-RS$ is a null matrix.