This stack focuses on the first and most crucial step of matrix multiplication: determining if two matrices can be multiplied and, if so, what the dimensions of the resulting product matrix will be. Mastering this prevents common errors.
What is the essential condition for multiplying two matrices, \(A\) and \(B\), to find the product \(AB\)?
The number of columns in the first matrix (\(A\)) must be equal to the number of rows in the second matrix (\(B\)).
If matrix \(A\) has dimensions \(m \times n\) and matrix \(B\) has dimensions \(p \times q\), what must be true to find the product \(AB\)?
The 'inner' dimensions must match: \(n\) must be equal to \(p\). So, \(n=p\).
If \(A\) is an \(m \times n\) matrix and \(B\) is an \(n \times p\) matrix, what are the dimensions of the product matrix \(AB\)?
The dimensions of the product matrix are the 'outer' dimensions: \(m \times p\).
Let \(A\) be a \(3 \times 4\) matrix and \(B\) be a \(4 \times 2\) matrix. Can the product \(AB\) be computed? If so, what are its dimensions?
Yes, it can be computed because the number of columns in \(A\) (\(4\)) equals the number of rows in \(B\) (\(4\)). The dimensions of \(AB\) will be \(3 \times 2\).
Let \(C\) be a \(2 \times 5\) matrix and \(D\) be a \(2 \times 3\) matrix. Can the product \(CD\) be computed?
No. The number of columns in \(C\) (\(5\)) does not equal the number of rows in \(D\) (\(2\)).
If matrix product \(AB\) is defined, does this mean that the product \(BA\) is also defined?
Not necessarily. For \(BA\) to be defined, the number of columns in \(B\) must equal the number of rows in \(A\).
Let \(A\) be a \(2 \times 3\) matrix and \(B\) be a \(3 imes 2\) matrix. What are the dimensions of the product \(AB\)?
The inner dimensions match (\(3\) and \(3\)), so the product \(AB\) is defined. Its dimensions are the outer dimensions: \(2 \times 2\).
Let \(A\) be a \(2 \times 3\) matrix and \(B\) be a \(3 \times 2\) matrix. What are the dimensions of the product \(BA\)?
The inner dimensions match (\(2\) and \(2\)), so the product \(BA\) is defined. Its dimensions are the outer dimensions: \(3 \times 3\).
For which type of matrices \(A\) and \(B\) are both \(AB\) and \(BA\) guaranteed to be defined and have the same dimensions?
This is guaranteed if both \(A\) and \(B\) are square matrices of the same size (e.g., both are \(n \times n\) matrices).
