1. **State the problem:** Given matrices $$A = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ -3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$$ we need to determine which of the following is true: (a) $AB$ is not defined (b) $AB = \begin{bmatrix} 11 \\ 6 \\ 6 \end{bmatrix}$ (c) None of the above 2. **Recall matrix multiplication rules:** - The product $AB$ is defined only if the number of columns of $A$ equals the number of rows of $B$. - $A$ is a $3 \times 2$ matrix (3 rows, 2 columns). - $B$ is a $2 \times 1$ matrix (2 rows, 1 column). - Since $A$ has 2 columns and $B$ has 2 rows, $AB$ is defined. 3. **Calculate $AB$:** $$AB = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ -3 & 4 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} (1)(2) + (-1)(3) \\ (0)(2) + (2)(3) \\ (-3)(2) + (4)(3) \end{bmatrix} = \begin{bmatrix} 2 - 3 \\ 0 + 6 \\ -6 + 12 \end{bmatrix} = \begin{bmatrix} -1 \\ 6 \\ 6 \end{bmatrix}$$ 4. **Compare with given options:** - Option (a) is false because $AB$ is defined. - Option (b) states $AB = \begin{bmatrix} 11 \\ 6 \\ 6 \end{bmatrix}$, which is incorrect. - Therefore, option (c) "None of the above is true" is correct. **Final answer:** (c) None of the above is true