Compute each individual multiplication within the matrix entries: 6 × 2 = 12, 2 × 7 = 14, 6 × 6 = 36, 2 × 9 = 18, 6 × 0 = 0, 2 × 8 = 16, 9 × 2 = 18, 3 × 7 = 21, 9 × 6 = 54, 3 × 9 = 27, and 3 × 8 = 24.

Compute each individual multiplication within the matrix entries: 6 × 2 = 12, 2 × 7 = 14, 6 × 6 = 36, 2 × 9 = 18, 6 × 0 = 0, 2 × 8 = 16, 9 × 2 = 18, 3 × 7 = 21, 9 × 6 = 54, 3 × 9 = 27, and 3 × 8 = 24.

If $A = \begin{bmatrix} 6 & 2 \\ 9 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$ , find $AB$ .

Answer: $AB = \begin{pmatrix} 26 & 54 & 16 \\ 39 & 81 & 24 \end{pmatrix}$

Step-by-step solution

Step 1: Check order and conformability

Matrix multiplication $AB$ is defined because the number of columns of matrix $A$ equals 2, which matches the number of rows of matrix $B$ , which is also 2. The resulting matrix product $AB$ will have dimension $2 \times 3$ .

Step 2: Set up matrix multiplication

Each entry $(i, j)$ in $AB$ is obtained by taking the dot product of row $i$ of matrix $A$ with column $j$ of matrix $B$ . We write out the sum of products for each of the 6 entries.

Step 3: Simplify intermediate products

Compute each individual multiplication within the matrix entries: $6 \times 2 = 12$ , $2 \times 7 = 14$ , $6 \times 6 = 36$ , $2 \times 9 = 18$ , $6 \times 0 = 0$ , $2 \times 8 = 16$ , $9 \times 2 = 18$ , $3 \times 7 = 21$ , $9 \times 6 = 54$ , $3 \times 9 = 27$ , and $3 \times 8 = 24$ .

Step 4: Calculate the final matrix

Add the terms inside each entry: $12 + 14 = 26$ , $36 + 18 = 54$ , $0 + 16 = 16$ , $18 + 21 = 39$ , $54 + 27 = 81$ , and $0 + 24 = 24$ . This yields the final product matrix.

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