Question

Assuming that the sums and products given below are defined, which of the following is not true for matrices
A. A + B = B + A
B. AB = AC does not implies B = C
C. AB = 0 implies A = 0 or B = 0
D. $(AB)^{\prime}=B^{\prime} A^{\prime}$

Answer 1

Hint: In this question we need to find the wrong option regarding the addition and multiplication of matrices. Therefore, to solve it is necessary to recall the properties of matrix addition and multiplication, and also try to find examples that contradict the statements.

Complete step by step solution: Let's go through each option.
Option A. says A + B = B + A. We know that the addition of matrices is commutative. Therefore, option A is correct about the addition of matrices.
In option B we have AB = AC we need to verify whether B = C or not. To verify this statement, we can consider two cases - first A is invertible and second A is not invertible.
If A is invertible, then $A^{-1}$ exists. We have,
AB = AC
Multiply both sides by $A^{-1}$ to the left side,
$A^{-1} \mathrm{AB}=A^{-1} \mathrm{AC}$
We know that since A is an invertible matrix, $A A^{-1}=1$. Therefore, we get B = C.
Now consider the second case - A is not invertible, then $A^{-1}$ does not exist. In this case, we consider an example to validate the statement given.
Let A be a null matrix, then AC = AB = 0 regardless of B and C. So, option B is correct.
To check whether option C is wrong or not we need to find an example that violates the statement.
Let $A = \begin{bmatrix}0 & 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix}1 & 0 & 0 \end{bmatrix}$, then AB = 0 but both A and B are non-zero matrices.So, option C is wrong.
Option D is the property of matrix transpose, therefore option D is correct.

Option ‘C’ is correct

Note: When solving these kinds of questions, always try to find examples that contradict the given statements. Here, multiplication means matrix multiplication does not compare the given statements with that of scalar multiplication, since the results that are true for scalar multiplication might be wrong for matrix multiplication.