Multiply matrices

Inner dimensions match; outer dimensions remain

If A has m rows and n columns, and B has n rows and p columns, then their product has m rows and p columns. The entry in row i and column j is the sum of the products of row i of A with column j of B. The sum has n terms

A has order 2×3 and B has order 3×4. How many columns does AB have?

Matrix dimensions

\(\displaystyle A:2\times3,\quad B:3\times4\)

For those dimensions, which statement about BA is correct?

One row and one column determine one entry

Use the matrices A and B. The first product entry comes from the first row of A and first column of B

1. 1First product entry Multiply matching entries, then add \(\displaystyle 1\times2+2\times(-1)+(-1)\times3=-3\)

Given matrix

\(\displaystyle \begin{gathered}A=\begin{bmatrix} 1 & 2 & -1 \\ 0 & 3 & 4 \end{bmatrix}\\B=\begin{bmatrix} 2 & 1 \\ -1 & 5 \\ 3 & 0 \end{bmatrix}\end{gathered}\)

For this A and B, find (AB)₂₁

Identity and powers use matrix multiplication

For a matrix with m rows and n columns, compatible identity matrices leave it unchanged. Squaring a matrix means multiplying it by itself and is defined only for a square matrix; it is not entrywise squaring. The zeroth power of a square matrix is its identity matrix. Matrix multiplication is associative and distributes over compatible sums, but generally does not commute

Given matrix

\(\displaystyle A=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)

For matrix A, find the top-right entry after multiplying A by itself

Give counterexamples, ungraded

Show that AB need not equal BA even when both products are defined, and that a product of two nonzero matrices can be zero. Independent construction and justification are ungraded

Model: compute both products and inspect the factors

For the displayed matrices, multiplying A by B gives a different result from multiplying B by A, so multiplication is not generally commutative. Squaring the nonzero matrix N gives the zero matrix, showing that a zero product does not force a zero factor. This also warns against cancellation without an invertibility hypothesis