Mathematics Multiple Choice Questions on “Operations on Matrices”.
1. The addition of matrices is only possible if they are of the same order.
a) True
b) False
Answer: a
Clarification: The given statement is true. Addition of matrices is possible only if the matrices are of the same order. If there are two matrices of different order, then A+B is not defined.
2. If A = (begin{bmatrix}1&2&3\9&10&11end{bmatrix}) and B = (begin{bmatrix}0&5&0\5&0&5end{bmatrix}), then find A+B.
a) A+B = (begin{bmatrix}1&7&3\11&10&16end{bmatrix})
b) A+B = (begin{bmatrix}1&7&3\14&11&13end{bmatrix})
c) A+B = (begin{bmatrix}1&7&3\14&10&16end{bmatrix})
d) A+B = (begin{bmatrix}1&5&3\14&10&16end{bmatrix})
Answer: c
Clarification: Given that, A = (begin{bmatrix}1&2&3\9&10&11end{bmatrix}) and B = (begin{bmatrix}0&5&0\5&0&5end{bmatrix})
Then A+B = (begin{bmatrix}1+0&2+5&3+0\9+5&10+0&11+5end{bmatrix}) = (begin{bmatrix}1&7&3\14&10&16end{bmatrix}).
3. If A = (begin{bmatrix}3&4\1&2end{bmatrix}) and B = (begin{bmatrix}1&5\2&3end{bmatrix}), find 2A-3B.
a) (begin{bmatrix}3&7\-4&5end{bmatrix})
b) (begin{bmatrix}-3&-7\-4&-5end{bmatrix})
c) (begin{bmatrix}3&7\-4&-5end{bmatrix})
d) (begin{bmatrix}3&-7\-4&-5end{bmatrix})
Answer: d
Clarification: Given that, A = (begin{bmatrix}3&4\1&2end{bmatrix}) and B = (begin{bmatrix}1&5\2&3end{bmatrix})
⇒2A=2(begin{bmatrix}3&4\1&2end{bmatrix})=(begin{bmatrix}6&8\2&4end{bmatrix}) and 3B=3(begin{bmatrix}1&5\2&3end{bmatrix})=(begin{bmatrix}3&15\6&9end{bmatrix})
∴2A-3B = (begin{bmatrix}6&8\2&4end{bmatrix})–(begin{bmatrix}3&15\6&9end{bmatrix})=(begin{bmatrix}3&-7\-4&-5end{bmatrix}).
4. If A+B = (begin{bmatrix}6&7\5&0end{bmatrix})and A = (begin{bmatrix}2&5\1&-1end{bmatrix}). Find the matrix B.
a) B = (begin{bmatrix}4&1\2&4end{bmatrix})
b) B = (begin{bmatrix}4&2\4&1end{bmatrix})
c) B = (begin{bmatrix}4&1\4&2end{bmatrix})
d) B = (begin{bmatrix}4&4\4&2end{bmatrix})
Answer: b
Clarification: Given that, A+B = (begin{bmatrix}6&7\5&0end{bmatrix})and A = (begin{bmatrix}2&5\1&-1end{bmatrix})
⇒B=(A+B)-A = (begin{bmatrix}6&7\5&0end{bmatrix})–(begin{bmatrix}2&5\1&-1end{bmatrix})
B = (begin{bmatrix}4&2\4&1end{bmatrix})
5. Find the matrix M and N, if M+N = (begin{bmatrix}5&6\7&8end{bmatrix}),M-N = (begin{bmatrix}4&5\6&8end{bmatrix}).
a) M=1/2 (begin{bmatrix}9&11\13&16end{bmatrix}), N=1/2 (begin{bmatrix}1&1\1&0end{bmatrix})
b) M=(begin{bmatrix}5&6\7&8end{bmatrix}), N=(begin{bmatrix}4&5\8&6end{bmatrix})
c) M=1/2 (begin{bmatrix}9&2\13&16end{bmatrix}), N=1/2 (begin{bmatrix}1&1\2&5end{bmatrix})
d) M=1/2 (begin{bmatrix}4&5\1&2end{bmatrix}), N=1/2 (begin{bmatrix}1&2\1&2end{bmatrix})
Answer: a
Clarification:M+N = (begin{bmatrix}5&6\7&8end{bmatrix})-(1) and M-N = (begin{bmatrix}4&5\6&8end{bmatrix})-(2)
Adding equation (1) and equation (2), (M+N)+(M-N)=2M=(begin{bmatrix}5&6\7&8end{bmatrix})+(begin{bmatrix}4&5\6&8end{bmatrix})
M=1/2 (begin{bmatrix}9&11\13&16end{bmatrix}).
Subtracting equation (1) and equation (2), (M+N)-(M-N)=2N=(begin{bmatrix}5&6\7&8end{bmatrix})–(begin{bmatrix}4&5\6&8end{bmatrix})
N=1/2 (begin{bmatrix}1&1\1&0end{bmatrix}).
6. Find the value of x and y if 2(begin{bmatrix}5&x\y-4&6end{bmatrix})+(begin{bmatrix}-4&1\3&2end{bmatrix})=(begin{bmatrix}6&3\10&14end{bmatrix})?
a) x=-1, y=9
b) x=-1, y=-9
c) x=1, y=-9
d) x=1, y=9
Answer: d
Clarification: Given that, 2(begin{bmatrix}5&x\y-4&6end{bmatrix})+(begin{bmatrix}-4&1\3&2end{bmatrix})=(begin{bmatrix}6&3\10&14end{bmatrix})
⇒(begin{bmatrix}2(5)-4&2x+1\2(y-4)+3&2(6)+2end{bmatrix})=(begin{bmatrix}6&3\10&14end{bmatrix})
Comparing the two matrices, 2x+1=3, 2y-8=10
Solving the two equations we get, x=1, y=9.
7. Find AB if A = (begin{bmatrix}1&2\3&4end{bmatrix}) and B = (begin{bmatrix}1&5\3&2end{bmatrix}).
a) AB = (begin{bmatrix}15&23\9&7end{bmatrix})
b) AB = (begin{bmatrix}9&7\23&15end{bmatrix})
c) AB = (begin{bmatrix}7&9\15&23end{bmatrix})
d) AB = (begin{bmatrix}7&9\23&15end{bmatrix})
Answer: c
Clarification: Given that, A = (begin{bmatrix}1&2\3&4end{bmatrix}) and B = (begin{bmatrix}1&5\3&2end{bmatrix})
Then, AB = (begin{bmatrix}1&2\3&4end{bmatrix})(begin{bmatrix}1&5\3&2end{bmatrix})
=(begin{bmatrix}1×1+2×3&1×5+2×2\3×1+4×3&3×5+4×2end{bmatrix})=(begin{bmatrix}7&9\15&23end{bmatrix}).
8. Matrix addition and matrix multiplication both are commutative.
a) True
b) False
Answer: b
Clarification: The given statement is false. Matrix addition is commutative i.e. A+B=B+A. But matrix multiplication is not commutative i.e.AB≠BA.
9. Let A=(begin{bmatrix}3&-5&2\-4&-6&2\7&1&5end{bmatrix}). Find the additive inverse of A.
a) (begin{bmatrix}-3&5&-2\-4&6&2\7&1&5end{bmatrix})
b) (begin{bmatrix}3&-5&2\-4&-6&2\7&1&5end{bmatrix})
c) (begin{bmatrix}-3&5&-2\4&6&-2\-7&-1&-5end{bmatrix})
d) (begin{bmatrix}-3&5&2\-4&6&-2\-7&-1&5end{bmatrix})
Answer: c
Clarification: Additive inverse of matrix A is the negative of A i.e. -A.
Therefore, -A=(begin{bmatrix}-3&5&-2\4&6&-2\-7&-1&-5end{bmatrix})
10. Which of the following condition is incorrect for matrix multiplication?
a) A(BC)=(AB)C
b) A(B+C)=AB+AC
c) AB=0 if either A or B is 0
d) AB=BA
Answer: d
Clarification: Matrix multiplication is never commutative i.e. AB≠BA. Therefore, the condition AB=BA is incorrect.