Question

If \[ A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \] are three matrices, then find $ABC$.

Hint:

When multiplying three matrices, compute them two at a time: first $AB$, then multiply the result with $C$.

Answer 1

The matrices provided are: \[ A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \] The first step is to compute the product $AB$. \[ AB = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix} \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix} \] Performing the matrix multiplication: \[ AB = \begin{bmatrix} 1 \cdot 2 + (-1) \cdot (-1) + 0 \cdot 0 & 1 \cdot 0 + (-1) \cdot 3 + 0 \cdot 5 & 1 \cdot 1 + (-1) \cdot 4 + 0 \cdot 1 \end{bmatrix} \] This simplifies to: \[ AB = \begin{bmatrix} 2 + 1 + 0 & 0 - 3 + 0 & 1 - 4 + 0 \end{bmatrix} = \begin{bmatrix} 3 & -3 & -3 \end{bmatrix} \] The second step is to compute $(AB)C$. \[ ABC = \begin{bmatrix} 3 & -3 & -3 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \] The result of this multiplication is: \[ ABC = 3 \cdot 2 + (-3) \cdot 3 + (-3) \cdot 4 = 6 - 9 - 12 = -15 \]