Matrices: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Give a $2\times 2$ skew-symmetric matrix.
2. Possible orders of a matrix with 7 elements.
3. If $\left[\begin{array}{cc}x& 2\\ 3& y\end{array}\right]=\left[\begin{array}{cc}5& 2\\ 3& -1\end{array}\right]$, find $x+y$.
4. Write $a_{23}$ of $[a_{ij}]$ with $a_{ij}=i+2j$.
5. If $A=[12]$, find $A{A}^{\prime }$.
6. Order of $A^{\prime }$ if A is $3\times 5$.
7. If A is skew, what is $A+A^{\prime }$?
8. If $A=\mathrm{diag}(2,3)$, find $A^2$.
9. Is $\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$ symmetric?
10. When is $A+B$ defined?

D. Short answer I (2 marks each)

1. Find $AB$ for $A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$, $B=\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$.
2. Find X if $2X+\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}3& 8\\ 7& 2\end{array}\right]$.
3. Show that $A=\left[\begin{array}{cc}0& 2\\ -2& 0\end{array}\right]$ is skew and find $A^2$.
4. If $A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$, show $A^2-4A+3I=O$.
5. Construct the $2\times 3$ matrix $a_{ij}=i\cdot j$.
6. Find $x$ if $[x1]\left[\begin{array}{cc}1& 0\\ -2& 0\end{array}\right]=O$.
7. Write $\left[\begin{array}{cc}2& 4\\ 0& 6\end{array}\right]$ as symmetric + skew.
8. Show that $A{A}^{\prime }$ is symmetric for any matrix A.

E. Short answer II (3 marks each)

1. If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, prove $A^n=2^{n-1}A$ for $n=2,3$.
2. Find $x,y,z,w$ if $\left[\begin{array}{cc}x-y& 2z+w\\ 2x-y& 2x+w\end{array}\right]=\left[\begin{array}{cc}-1& 5\\ 0& 10\end{array}\right]$.
3. For $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, $B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, verify $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$.
4. If A and B are symmetric, show $AB$ is symmetric iff $AB=BA$.
5. If $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, show $(aI+bA{)}^2=a^{2}I+2abA$.
6. A shop sells 3 pens and 2 notebooks to A, 1 pen and 4 notebooks to B at ₹10 and ₹25. Find the bills by matrix multiplication.

F. Long answer (5 marks each)

1. (a) Express $\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix. (b) Show that it is idempotent ($A^2=A$).
2. (a) If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$, prove $A^2=\left[\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{array}\right]$. (b) Find $k$ if $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ satisfies $A^2=5A+kI$.

G. HOTS (higher-order thinking)

1. Can a non-zero matrix satisfy $A^2=O$? Give one.
2. If A is skew-symmetric, is $A^2$ symmetric or skew?
3. How many $2\times 2$ symmetric matrices have entries 0 or 1?
4. If AB and BA are both defined, what can you say about the orders of A and B?

20 MCQs

1. Order of AB, A $2\times 3$, B $3\times 4$
   (a) $3\times 3$ (b) $4\times 2$ (c) $2\times 4$ (d) not defined
2. Number of orders for 12 elements
   (a) 6 (b) 4 (c) 3 (d) 12
3. Skew-symmetric diagonal entries
   (a) 1 (b) equal (c) any (d) 0
4. $(A+B)(A-B)=A^2-B^2$ if
   (a) always (b) $AB=BA$ (c) $A=B$ (d) never
5. $(ABC{)}^{\prime }=$
   (a) $A^{\prime }{B}^{\prime }{C}^{\prime }$ (b) $C^{\prime }{B}^{\prime }{A}^{\prime }$ (c) $B^{\prime }{C}^{\prime }{A}^{\prime }$ (d) $CBA$
6. A matrix both symmetric and skew is
   (a) identity (b) scalar (c) diagonal (d) zero
7. $A^2=A$, then $(I-A{)}^2=$
   (a) $I-A$ (b) $I+A$ (c) $A$ (d) $O$
8. $\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$ is
   (a) skew (b) zero (c) scalar (d) row
9. $3\times 3$ matrices with entries 0, 1, 2
   (a) 27 (b) $2^9$ (c) 81 (d) $3^9$
10. For symmetric A, $a_{12}=5$, then $a_{21}=$
    (a) $-5$ (b) 5 (c) 0 (d) 25
11. If A is $m\times n$ and $AB$, $BA$ both exist, B is
    (a) $m\times n$ (b) $m\times m$ (c) $n\times m$ (d) $n\times n$
12. $\left[\begin{array}{cc}x& 0\\ 0& y\end{array}\right]=I$ gives $x+y=$
    (a) 2 (b) 0 (c) 1 (d) $-2$
13. For skew A, B: $AB+BA$ is
    (a) skew (b) symmetric (c) zero (d) identity
14. $I^n$ equals
    (a) $nI$ (b) $O$ (c) $I$ (d) $2I$
15. If $A=[a_{ij}{]}_{2\times 2}$, $a_{ij}=i+j$, then $a_{12}=$
    (a) 2 (b) 1 (c) 4 (d) 3
16. A column matrix has order
    (a) $m\times 1$ (b) $1\times n$ (c) $n\times n$ (d) $1\times 1$ only
17. $A$ symmetric, then $A^2$ is
    (a) symmetric (b) skew (c) zero (d) not defined
18. If $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$, $A^2=$
    (a) $-I$ (b) $O$ (c) $A$ (d) $I$
19. $\frac{1}{2}(A-A^{\prime })$ is
    (a) symmetric (b) skew-symmetric (c) diagonal (d) identity
20. $(A+B{)}^{\prime }=$
    (a) $A^{\prime }{B}^{\prime }$ (b) $B^{\prime }{A}^{\prime }$ (c) $A^{\prime }+B^{\prime }$ (d) $A+B$

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: For any square A, $A+A^{\prime }$ is symmetric. R: $(A+A^{\prime }{)}^{\prime }=A^{\prime }+A$.
2. A: $\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$ is a scalar matrix. R: a scalar matrix is a diagonal matrix.
3. A: If A is $2\times 3$, $A^{\prime }A$ is $3\times 3$. R: $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$.
4. A: $AB=O$ forces $A=O$ or $B=O$. R: matrix multiplication is not commutative.
5. A: Every square matrix is the sum of a symmetric and a skew matrix. R: $A=\frac{1}{2}(A+A^{\prime })+\frac{1}{2}(A-A^{\prime })$.
6. A: $\left[\begin{array}{cc}0& 3\\ -3& 0\end{array}\right]$ is skew-symmetric. R: its transpose equals its negative.

Key: 1 (a) · 2 (d) · 3 (b) · 4 (d) · 5 (a) · 6 (a)

Answers to the practice set