To find the product of the given matrices, we will perform matrix multiplication. Let's denote the first matrix (let's call it \(A\)) and the second matrix (let's call it \(B\)):
\[ A = \begin{bmatrix} 0 & -1 & 0 \ -5 & -2 & -5 \ 0 & 0 & 4 \end{bmatrix} \]
\[ B = \begin{bmatrix} 1 & 0 & 1 \ -1 & 0 & -1 \ 1 & -1 & 0 \end{bmatrix} \]
Step 1: Dimension Check
Matrix \(A\) is of size \(3 \times 3\) and matrix \(B\) is also of size \(3 \times 3\). Hence, the resulting product \(C = A \times B\) will also be of size \(3 \times 3\).
Step 2: Perform Multiplication
The entry in the resulting matrix \(C\) at position \(C[i][j]\) is computed as follows:
\[ C[i][j] = \sum_{k=1}^{3} A[i][k] \cdot B[k][j] \]
Let's calculate each entry of the resulting matrix \(C\):
For \(C[1][1]\):
\[ C[1][1] = A[1][1] \cdot B[1][1] + A[1][2] \cdot B[2][1] + A[1][3] \cdot B[3][1] \] \[ = 0 \cdot 1 + (-1) \cdot (-1) + 0 \cdot 1 = 0 + 1 + 0 = 1 \]
For \(C[1][2]\):
\[ C[1][2] = A[1][1] \cdot B[1][2] + A[1][2] \cdot B[2][2] + A[1][3] \cdot B[3][2] \] \[ = 0 \cdot 0 + (-1) \cdot 0 + 0 \cdot (-1) = 0 + 0 + 0 = 0 \]
For \(C[1][3]\):
\[ C[1][3] = A[1][1] \cdot B[1][3] + A[1][2] \cdot B[2][3] + A[1][3] \cdot B[3][3] \] \[ = 0 \cdot 1 + (-1) \cdot (-1) + 0 \cdot 0 = 0 + 1 + 0 = 1 \]
For \(C[2][1]\):
\[ C[2][1] = A[2][1] \cdot B[1][1] + A[2][2] \cdot B[2][1] + A[2][3] \cdot B[3][1] \] \[ = (-5) \cdot 1 + (-2) \cdot (-1) + (-5) \cdot 1 = -5 + 2 - 5 = -8 \]
For \(C[2][2]\):
\[ C[2][2] = A[2][1] \cdot B[1][2] + A[2][2] \cdot B[2][2] + A[2][3] \cdot B[3][2] \] \[ = (-5) \cdot 0 + (-2) \cdot 0 + (-5) \cdot (-1) = 0 + 0 + 5 = 5 \]
For \(C[2][3]\):
\[ C[2][3] = A[2][1] \cdot B[1][3] + A[2][2] \cdot B[2][3] + A[2][3] \cdot B[3][3] \] \[ = (-5) \cdot 1 + (-2) \cdot (-1) + (-5) \cdot 0 = -5 + 2 + 0 = -3 \]
For \(C[3][1]\):
\[ C[3][1] = A[3][1] \cdot B[1][1] + A[3][2] \cdot B[2][1] + A[3][3] \cdot B[3][1] \] \[ = 0 \cdot 1 + 0 \cdot (-1) + 4 \cdot 1 = 0 + 0 + 4 = 4 \]
For \(C[3][2]\):
\[ C[3][2] = A[3][1] \cdot B[1][2] + A[3][2] \cdot B[2][2] + A[3][3] \cdot B[3][2] \] \[ = 0 \cdot 0 + 0 \cdot 0 + 4 \cdot (-1) = 0 + 0 - 4 = -4 \]
For \(C[3][3]\):
\[ C[3][3] = A[3][1] \cdot B[1][3] + A[3][2] \cdot B[2][3] + A[3][3] \cdot B[3][3] \] \[ = 0 \cdot 1 + 0 \cdot (-1) + 4 \cdot 0 = 0 + 0 + 0 = 0 \]
Step 3: Construct the Result Matrix
Combining all of the results, we get the resulting matrix \(C\):
\[ C = \begin{bmatrix} 1 & 0 & 1 \ -8 & 5 & -3 \ 4 & -4 & 0 \end{bmatrix} \]
Final Answer:
The product of the matrices \(A\) and \(B\) is:
\[ \begin{bmatrix} 1 & 0 & 1 \ -8 & 5 & -3 \ 4 & -4 & 0 \end{bmatrix} \]