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Math  /  Algebra

QuestionASK YOUR TE
If possible, find ABA B. (If not possible, enter IMPOSSIBLE in any cell of the matrix.) A=[0−126035−16],B=[4−14−516]A=\left[\begin{array}{rrr} 0 & -1 & 2 \\ 6 & 0 & 3 \\ 5 & -1 & 6 \end{array}\right], \quad B=\left[\begin{array}{rr} 4 & -1 \\ 4 & -5 \\ 1 & 6 \end{array}\right] AB=A B= □\square □\square □\square □\square □\square □\square - 1^\hat{1}
State the dimension of the result. (If not possible, enter IMPOSSIBLE in both answer blanks.) □\square ×\times □\square

Studdy Solution

STEP 1

1. Matrix multiplication is defined when the number of columns in the first matrix equals the number of rows in the second matrix.
2. The resulting matrix will have dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix.

STEP 2

1. Verify if matrix multiplication is possible.
2. Determine the dimensions of the resulting matrix.
3. Calculate the elements of the resulting matrix.

STEP 3

First, check if the multiplication of matrices A A and B B is possible. Matrix A A has dimensions 3×3 3 \times 3 , and matrix B B has dimensions 3×2 3 \times 2 . Since the number of columns in A A (3) is equal to the number of rows in B B (3), matrix multiplication is possible.

STEP 4

Determine the dimensions of the resulting matrix AB AB . The resulting matrix will have the number of rows of A A and the number of columns of B B . Therefore, the dimensions of AB AB are 3×2 3 \times 2 .

STEP 5

Calculate each element of the resulting matrix AB AB using the formula for matrix multiplication: (AB)ij=∑k=1nAik⋅Bkj (AB)_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj}
Calculate each element:
1. First row, first column: $ (AB)_{11} = (0 \cdot 4) + (-1 \cdot 4) + (2 \cdot 1) = 0 - 4 + 2 = -2 \]
2. First row, second column: $ (AB)_{12} = (0 \cdot -1) + (-1 \cdot -5) + (2 \cdot 6) = 0 + 5 + 12 = 17 \]
3. Second row, first column: $ (AB)_{21} = (6 \cdot 4) + (0 \cdot 4) + (3 \cdot 1) = 24 + 0 + 3 = 27 \]
4. Second row, second column: $ (AB)_{22} = (6 \cdot -1) + (0 \cdot -5) + (3 \cdot 6) = -6 + 0 + 18 = 12 \]
5. Third row, first column: $ (AB)_{31} = (5 \cdot 4) + (-1 \cdot 4) + (6 \cdot 1) = 20 - 4 + 6 = 22 \]
6. Third row, second column: $ (AB)_{32} = (5 \cdot -1) + (-1 \cdot -5) + (6 \cdot 6) = -5 + 5 + 36 = 36 \]
Thus, the resulting matrix AB AB is: AB=[−21727122236]AB = \begin{bmatrix} -2 & 17 \\ 27 & 12 \\ 22 & 36 \end{bmatrix}
The dimension of the resulting matrix is 3×2 3 \times 2 .

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