# Mathematics Questions and Answers – Operations on Matrices

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Operations on Matrices”.

1. The addition of matrices is only possible if they are of the same order.
a) True
b) False

Explanation: 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 = $$\begin{bmatrix}1&2&3\\9&10&11\end{bmatrix}$$ and B = $$\begin{bmatrix}0&5&0\\5&0&5\end{bmatrix}$$, then find A+B.
a) A+B = $$\begin{bmatrix}1&7&3\\11&10&16\end{bmatrix}$$
b) A+B = $$\begin{bmatrix}1&7&3\\14&11&13\end{bmatrix}$$
c) A+B = $$\begin{bmatrix}1&7&3\\14&10&16\end{bmatrix}$$
d) A+B = $$\begin{bmatrix}1&5&3\\14&10&16\end{bmatrix}$$

Explanation: Given that, A = $$\begin{bmatrix}1&2&3\\9&10&11\end{bmatrix}$$ and B = $$\begin{bmatrix}0&5&0\\5&0&5\end{bmatrix}$$
Then A+B = $$\begin{bmatrix}1+0&2+5&3+0\\9+5&10+0&11+5\end{bmatrix}$$ = $$\begin{bmatrix}1&7&3\\14&10&16\end{bmatrix}$$.

3. If A = $$\begin{bmatrix}3&4\\1&2\end{bmatrix}$$ and B = $$\begin{bmatrix}1&5\\2&3\end{bmatrix}$$, find 2A-3B.
a) $$\begin{bmatrix}3&7\\-4&5\end{bmatrix}$$
b) $$\begin{bmatrix}-3&-7\\-4&-5\end{bmatrix}$$
c) $$\begin{bmatrix}3&7\\-4&-5\end{bmatrix}$$
d) $$\begin{bmatrix}3&-7\\-4&-5\end{bmatrix}$$

Explanation: Given that, A = $$\begin{bmatrix}3&4\\1&2\end{bmatrix}$$ and B = $$\begin{bmatrix}1&5\\2&3\end{bmatrix}$$
⇒2A=2$$\begin{bmatrix}3&4\\1&2\end{bmatrix}$$=$$\begin{bmatrix}6&8\\2&4\end{bmatrix}$$ and 3B=3$$\begin{bmatrix}1&5\\2&3\end{bmatrix}$$=$$\begin{bmatrix}3&15\\6&9\end{bmatrix}$$
∴2A-3B = $$\begin{bmatrix}6&8\\2&4\end{bmatrix}$$–$$\begin{bmatrix}3&15\\6&9\end{bmatrix}$$=$$\begin{bmatrix}3&-7\\-4&-5\end{bmatrix}$$.
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4. If A+B = $$\begin{bmatrix}6&7\\5&0\end{bmatrix}$$and A = $$\begin{bmatrix}2&5\\1&-1\end{bmatrix}$$. Find the matrix B.
a) B = $$\begin{bmatrix}4&1\\2&4\end{bmatrix}$$
b) B = $$\begin{bmatrix}4&2\\4&1\end{bmatrix}$$
c) B = $$\begin{bmatrix}4&1\\4&2\end{bmatrix}$$
d) B = $$\begin{bmatrix}4&4\\4&2\end{bmatrix}$$

Explanation: Given that, A+B = $$\begin{bmatrix}6&7\\5&0\end{bmatrix}$$and A = $$\begin{bmatrix}2&5\\1&-1\end{bmatrix}$$
⇒B=(A+B)-A = $$\begin{bmatrix}6&7\\5&0\end{bmatrix}$$–$$\begin{bmatrix}2&5\\1&-1\end{bmatrix}$$
B = $$\begin{bmatrix}4&2\\4&1\end{bmatrix}$$

5. Find the matrix M and N, if M+N = $$\begin{bmatrix}5&6\\7&8\end{bmatrix}$$,M-N = $$\begin{bmatrix}4&5\\6&8\end{bmatrix}$$.
a) M=1/2 $$\begin{bmatrix}9&11\\13&16\end{bmatrix}$$, N=1/2 $$\begin{bmatrix}1&1\\1&0\end{bmatrix}$$
b) M=$$\begin{bmatrix}5&6\\7&8\end{bmatrix}$$, N=$$\begin{bmatrix}4&5\\8&6\end{bmatrix}$$
c) M=1/2 $$\begin{bmatrix}9&2\\13&16\end{bmatrix}$$, N=1/2 $$\begin{bmatrix}1&1\\2&5\end{bmatrix}$$
d) M=1/2 $$\begin{bmatrix}4&5\\1&2\end{bmatrix}$$, N=1/2 $$\begin{bmatrix}1&2\\1&2\end{bmatrix}$$

Explanation:M+N = $$\begin{bmatrix}5&6\\7&8\end{bmatrix}$$-(1) and M-N = $$\begin{bmatrix}4&5\\6&8\end{bmatrix}$$-(2)
Adding equation (1) and equation (2), (M+N)+(M-N)=2M=$$\begin{bmatrix}5&6\\7&8\end{bmatrix}$$+$$\begin{bmatrix}4&5\\6&8\end{bmatrix}$$
M=1/2 $$\begin{bmatrix}9&11\\13&16\end{bmatrix}$$.
Subtracting equation (1) and equation (2), (M+N)-(M-N)=2N=$$\begin{bmatrix}5&6\\7&8\end{bmatrix}$$–$$\begin{bmatrix}4&5\\6&8\end{bmatrix}$$
N=1/2 $$\begin{bmatrix}1&1\\1&0\end{bmatrix}$$.

6. Find the value of x and y if 2$$\begin{bmatrix}5&x\\y-4&6\end{bmatrix}$$+$$\begin{bmatrix}-4&1\\3&2\end{bmatrix}$$=$$\begin{bmatrix}6&3\\10&14\end{bmatrix}$$?
a) x=-1, y=9
b) x=-1, y=-9
c) x=1, y=-9
d) x=1, y=9

Explanation: Given that, 2$$\begin{bmatrix}5&x\\y-4&6\end{bmatrix}$$+$$\begin{bmatrix}-4&1\\3&2\end{bmatrix}$$=$$\begin{bmatrix}6&3\\10&14\end{bmatrix}$$
⇒$$\begin{bmatrix}2(5)-4&2x+1\\2(y-4)+3&2(6)+2\end{bmatrix}$$=$$\begin{bmatrix}6&3\\10&14\end{bmatrix}$$
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 = $$\begin{bmatrix}1&2\\3&4\end{bmatrix}$$ and B = $$\begin{bmatrix}1&5\\3&2\end{bmatrix}$$.
a) AB = $$\begin{bmatrix}15&23\\9&7\end{bmatrix}$$
b) AB = $$\begin{bmatrix}9&7\\23&15\end{bmatrix}$$
c) AB = $$\begin{bmatrix}7&9\\15&23\end{bmatrix}$$
d) AB = $$\begin{bmatrix}7&9\\23&15\end{bmatrix}$$

Explanation: Given that, A = $$\begin{bmatrix}1&2\\3&4\end{bmatrix}$$ and B = $$\begin{bmatrix}1&5\\3&2\end{bmatrix}$$
Then, AB = $$\begin{bmatrix}1&2\\3&4\end{bmatrix}$$$$\begin{bmatrix}1&5\\3&2\end{bmatrix}$$
=$$\begin{bmatrix}1×1+2×3&1×5+2×2\\3×1+4×3&3×5+4×2\end{bmatrix}$$=$$\begin{bmatrix}7&9\\15&23\end{bmatrix}$$.

8. Matrix addition and matrix multiplication both are commutative.
a) True
b) False

Explanation: 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=$$\begin{bmatrix}3&-5&2\\-4&-6&2\\7&1&5\end{bmatrix}$$. Find the additive inverse of A.
a) $$\begin{bmatrix}-3&5&-2\\-4&6&2\\7&1&5\end{bmatrix}$$
b) $$\begin{bmatrix}3&-5&2\\-4&-6&2\\7&1&5\end{bmatrix}$$
c) $$\begin{bmatrix}-3&5&-2\\4&6&-2\\-7&-1&-5\end{bmatrix}$$
d) $$\begin{bmatrix}-3&5&2\\-4&6&-2\\-7&-1&5\end{bmatrix}$$

Explanation: Additive inverse of matrix A is the negative of A i.e. -A.
Therefore, -A=$$\begin{bmatrix}-3&5&-2\\4&6&-2\\-7&-1&-5\end{bmatrix}$$

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

Explanation: Matrix multiplication is never commutative i.e. AB≠BA. Therefore, the condition AB=BA is incorrect.

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