Example Question - matrix product
Here are examples of questions we've helped users solve.
Matrix Multiplication with Identity Matrix
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: \[ A \cdot I_3 = \begin{bmatrix} 3 & -5 & 6 \\ -2 & 4 & 2 \\ -1 & 0 & 3 \end{bmatrix} \]
Matrix Multiplication Rules
For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix A is a 3x3 matrix, which means it has 3 rows and 3 columns. Matrix B is a 4x3 matrix, which means it has 4 rows and 3 columns. To multiply AB, the number of columns in A must equal the number of rows in B. Since A has 3 columns and B has 4 rows, AB cannot be calculated. To multiply BA, the number of columns in B must equal the number of rows in A. Since B has 3 columns and A has 3 rows, BA can be calculated. So, the correct answer is: B. only BA
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