The image shows a matrix equation where matrix A is given, and we need to calculate the product of matrix A with another matrix. Matrix A is: \[ A = \begin{bmatrix} 3 & -5 & 6 \\ -2 & 4 & 2 \\ -1 & 0 & 3 \end{bmatrix} \] And it needs to be multiplied by the matrix: \[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] This second matrix is the identity matrix \( I_3 \) of size 3x3. The matrix product of any matrix with the identity matrix of the appropriate size is the original matrix itself. That's because the identity matrix acts like the number 1 for matrix multiplication. Therefore, the product of matrix A with this identity matrix is matrix A unchanged: \[ A \cdot I_3 = \begin{bmatrix} 3 & -5 & 6 \\ -2 & 4 & 2 \\ -1 & 0 & 3 \end{bmatrix} \]
The image shows the handwritten task of calculating the matrix product A * I, where matrix A is a 3x3 matrix with entries: \[ A = \begin{bmatrix} 3 & -5 & 6 \\ 2 & 4 & 2 \\ 7 & 0 & 3 \end{bmatrix} \] and I is the 3x3 identity matrix, which is: \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] Multiplying any matrix by the identity matrix yields the original matrix. So the product A * I is just the matrix A itself: \[ A * I = A = \begin{bmatrix} 3 & -5 & 6 \\ 2 & 4 & 2 \\ 7 & 0 & 3 \end{bmatrix} \]
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