Question

To find the product \(A B\) of two matrices \(A\) and \(B,\) which statement must be true? (a) The number of columns in \(A\) must equal the number of rows in \(B\). (b) The number of rows in \(A\) must equal the number of columns in \(B\). (c) \(A\) and \(B\) must have the same number of rows and the same number of columns. (d) \(A\) and \(B\) must both be square matrices.

Short answer

Option (a) is true: The number of columns in \(A\) must equal the number of rows in \(B\).

Step-by-step solution

Understand Matrix Multiplication

Matrix multiplication is defined only when the number of columns in the first matrix equals the number of rows in the second matrix.

Analyze Each Option

Evaluate each given option:(a) The number of columns in \(A\) must equal the number of rows in \(B\).(b) The number of rows in \(A\) must equal the number of columns in \(B\).(c) \(A\) and \(B\) must have the same number of rows and the same number of columns.(d) \(A\) and \(B\) must both be square matrices.

Eliminate Incorrect Options

Options (c) and (d) involve conditions that are not necessary for matrix multiplication, i.e., having the same dimensions or both being square matrices.

Identify Necessary Condition

Option (a) states the correct requirement for matrix multiplication: the number of columns in \(A\) must equal the number of rows in \(B\). This is the correct condition for the product \(AB\) to be defined.

Key concepts

Matrix Product

When we talk about the matrix product, we refer to the result of multiplying two matrices together. This operation is fundamental in linear algebra.
To multiply two matrices, we perform a series of dot products between the rows of the first matrix (let's call it A) and the columns of the second matrix (B).
Each element in the resulting product matrix is the sum of these dot products.
For instance, to find the element in the first row and first column of the product matrix (which we can call C), we multiply each element of the first row of A by the corresponding element of the first column of B, and then sum these products.

A simple example: if \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\), the product \(AB = C\) is calculated as follows:

• The element in the first row and first column of C is \(ae + bg\)
• The element in the first row and second column of C is \(af + bh\)
• The element in the second row and first column of C is \(ce + dg\)
• The element in the second row and second column of C is \(cf + dh\)

Understanding this process helps you see how each element in the resulting matrix C is determined by specific elements from matrices A and B. This method ensures that we systematically compute each value in the product matrix.

Dimensions in Matrices

Dimensions play a critical role in matrix multiplication. Each matrix has dimensions that are defined as 'rows x columns'. For example, a matrix with three rows and two columns has dimensions \(3 \times 2\).

It is essential to understand that for two matrices to be multiplied, the number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B).
That is to say, if A has dimensions \(m \times n\), and B has dimensions \(p \times q\), then for their product to be defined, we must have \(n = p\).


This often forms a key condition in determining whether matrix multiplication is possible.

When this condition is fulfilled, the resulting matrix (called the product matrix) will have dimensions \(m \times q\).
For example, if matrix A is \(2 \times 3\) and matrix B is \(3 \times 4\), their product matrix will be \(2 \times 4\).
Ensuring compatibility of dimensions is crucial, as attempting to multiply matrices where this condition is not met will lead to undefined results.

Matrix Multiplication Rules

Matrix multiplication follows specific rules that must be adhered to for the operation to yield correct results.

• The number of columns in the first matrix must equal the number of rows in the second matrix. Without this alignment, you cannot perform the multiplication.
• Matrix multiplication is associative. This means that if you have three matrices A, B, and C, the order in which you multiply them does not matter: \((AB)C = A(BC)\).
• Matrix multiplication is distributive. This means that for three matrices A, B, and C: \(A(B + C) = AB + AC\).
• Matrix multiplication is generally not commutative. That is, \(AB\) does not necessarily equal \(BA\). The order in which you multiply matrices affects the result.

By adhering to these rules and the condition on the dimensions of the matrices, we can carry out matrix multiplication efficiently and accurately. Understanding these foundational principles ensures that we avoid common pitfalls and correctly apply the matrix product across various applications in linear algebra and beyond.