If $A = \begin{bmatrix}2&1&3\\4&-3&5\end{bmatrix}$ and $B = \begin{bmatrix}-2&3\\4&-5\\1&2\end{bmatrix}$, then which of the following statements are TRUE?

(A) AB is defined
(B) AB and BA both are defined and AB = I, where I is an identity matrix of order 2
(C) BA is defined
(D) AB and BA both are defined and AB = BA

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (3) → (A) and (C) only

Given matrices:

\[ A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -3 & 5 \end{bmatrix} \quad (2 \times 3) \]

\[ B = \begin{bmatrix} -2 & 3 \\ 4 & -5 \\ 1 & 2 \end{bmatrix} \quad (3 \times 2) \]

Check which products are defined:

• (A) \(AB\) is defined if number of columns of \(A\) equals number of rows of \(B\):

  Number of columns of \(A\) = 3, number of rows of \(B\) = 3 ⇒ \(AB\) is defined.

• (C) \(BA\) is defined if number of columns of \(B\) equals number of rows of \(A\):

  Number of columns of \(B\) = 2, number of rows of \(A\) = 2 ⇒ \(BA\) is defined.

Dimensions of products:

• \(AB\) will be a \(2 \times 2\) matrix.
• \(BA\) will be a \(3 \times 3\) matrix.

Calculate \(AB\):

\[ AB = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -3 & 5 \end{bmatrix} \times \begin{bmatrix} -2 & 3 \\ 4 & -5 \\ 1 & 2 \end{bmatrix} = ? \]

Calculate each element:

• \((AB)_{11} = 2 \times (-2) + 1 \times 4 + 3 \times 1 = -4 + 4 + 3 = 3\)
• \((AB)_{12} = 2 \times 3 + 1 \times (-5) + 3 \times 2 = 6 - 5 + 6 = 7\)
• \((AB)_{21} = 4 \times (-2) + (-3) \times 4 + 5 \times 1 = -8 -12 + 5 = -15\)
• \((AB)_{22} = 4 \times 3 + (-3) \times (-5) + 5 \times 2 = 12 + 15 + 10 = 37\)

\[ AB = \begin{bmatrix} 3 & 7 \\ -15 & 37 \end{bmatrix} \]

Since \(AB \neq I\), option (B) is false.

Check if \(AB = BA\):

Since \(AB\) is \(2 \times 2\) and \(BA\) is \(3 \times 3\), they cannot be equal.

Therefore, option (D) is false.

Summary:

• (A) True
• (B) False
• (C) True
• (D) False