Math  /  Algebra

QuestionASK YOUR TE
If possible, find ABA B. (If not possible, enter IMPOSSIBLE in any cell of the matrix.) A=[012603516],B=[414516]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
Calculate each element of the resulting matrix AB AB using the formula for matrix multiplication: (AB)ij=k=1nAikBkj (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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