Matrix Multiplication

How to multiply two matrices

Matrix multiplication falls into two general categories:

Scalar in which a single number is multiplied with every entry of a matrix
Multiplication of an entire matrix by another entire matrix For the rest of the page, matrix multiplication will refer to this second category.

Scalar Matrix Multiplication

In the scalar variety, every entry is multiplied by a number, called a scalar.
scalar matrix multipication picture

What is the answer to the scalar multiplication problem below?
scalar matrix multiplication example problem

Answer

(See how this problem can be represented as a Scalar Dilation)

Matrix Multiplication

You can multiply two matrices if, and only if, the number of columns in the first matrix equals the number of rows in the second matrix. Otherwise, the product of two matrices is undefined.

The product matrix's dimensions are

(rows of first matrix) × (columns of the second matrix )

In the picture on the left, the matrices can be multiplied since the number of columns in the 1st one, matrix A, equals the number of rows in the 2^nd, matrix B.

The Dimensions of the product matrix

Rows of 1^st matrix × Columns of 2^nd

4 × 3

Generalized Example
If we multiply a 2×3 matrix with a 3×1 matrix, the product matrix is 2×1

Here is how we get M₁₁ and M₂₂ in the product.

M₁₁ = r₁₁× t₁₁ + r₁₂× t₂₁ + r₁₃×t₃₁
M₁₂ = r₂₁× t₁₁ + r₂₂× t₂₁ + r₂₃×t₃₁

Matrix C and D below cannot be multiplied together because the number of columns in C does not equal the number of rows in D. In this case, the multiplication of these two matrices is not defined.
Matrix Multiply

How To Multiply Matrices

In order to multiply matrices,

Step 1: Make sure that the the number of columns in the 1^st one equals the number of rows in the 2^nd one. (The pre-requisite to be able to multiply)
Step 2: Multiply the elements of each row of the first matrix by the elements of each column in the second matrix.
Step 3: Add the products.

Is the product of matrix A and Matrix B below defined or undefined?

Answer

when multiplication is defined for matrices

Practice Problem

Multiply matrix A and B below to find the product.

Show Product

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