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For the matrices A and B, verify that (A...

For the matrices A and B, verify that (AB)′=B′A′, where (i) `A=[{:(-1),(4),(3):}],` `B=[-1" " 2" " 1]`

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To verify that \((AB)′ = B′A′\) for the matrices \(A\) and \(B\), we will follow these steps: ### Step 1: Define the matrices A and B Given: \[ A = \begin{bmatrix} -1 \\ 4 \\ 3 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix} \] ### Step 2: Calculate the product \(AB\) To find \(AB\), we multiply matrix \(A\) (which is \(3 \times 1\)) with matrix \(B\) (which is \(1 \times 3\)). The result will be a \(3 \times 3\) matrix. \[ AB = \begin{bmatrix} -1 \\ 4 \\ 3 \end{bmatrix} \begin{bmatrix} -1 & 2 & 1 \end{bmatrix} = \begin{bmatrix} (-1)(-1) & (-1)(2) & (-1)(1) \\ (4)(-1) & (4)(2) & (4)(1) \\ (3)(-1) & (3)(2) & (3)(1) \end{bmatrix} \] Calculating the elements: \[ AB = \begin{bmatrix} 1 & -2 & -1 \\ -4 & 8 & 4 \\ -3 & 6 & 3 \end{bmatrix} \] ### Step 3: Calculate the transpose of \(AB\) Now, we find \((AB)′\) by transposing the matrix \(AB\). \[ (AB)′ = \begin{bmatrix} 1 & -4 & -3 \\ -2 & 8 & 6 \\ -1 & 4 & 3 \end{bmatrix} \] ### Step 4: Calculate \(B′\) and \(A′\) Now, we need to find \(B′\) (the transpose of \(B\)) and \(A′\) (the transpose of \(A\)). \[ B′ = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix}, \quad A′ = \begin{bmatrix} -1 & 4 & 3 \end{bmatrix} \] ### Step 5: Calculate the product \(B′A′\) Now, we multiply \(B′\) (which is \(3 \times 1\)) with \(A′\) (which is \(1 \times 3\)). The result will also be a \(3 \times 3\) matrix. \[ B′A′ = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix} \begin{bmatrix} -1 & 4 & 3 \end{bmatrix} = \begin{bmatrix} (-1)(-1) & (-1)(4) & (-1)(3) \\ (2)(-1) & (2)(4) & (2)(3) \\ (1)(-1) & (1)(4) & (1)(3) \end{bmatrix} \] Calculating the elements: \[ B′A′ = \begin{bmatrix} 1 & -4 & -3 \\ -2 & 8 & 6 \\ -1 & 4 & 3 \end{bmatrix} \] ### Step 6: Compare \((AB)′\) and \(B′A′\) Now we compare the two results: \[ (AB)′ = \begin{bmatrix} 1 & -4 & -3 \\ -2 & 8 & 6 \\ -1 & 4 & 3 \end{bmatrix}, \quad B′A′ = \begin{bmatrix} 1 & -4 & -3 \\ -2 & 8 & 6 \\ -1 & 4 & 3 \end{bmatrix} \] Since both matrices are equal, we conclude that: \[ (AB)′ = B′A′ \] ### Conclusion Thus, we have verified that \((AB)′ = B′A′\). ---
To verify that \((AB)′ = B′A′\) for the matrices \(A\) and \(B\), we will follow these steps: ### Step 1: Define the matrices A and B Given: \[ A = \begin{bmatrix} -1 \\ 4 \\ 3 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix} \] ...
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