If A is a 2 × 3 matrix and AB is a 2 × 5 matrix, then B must be a
3 × 5 matrix
Understanding Matrix Dimensions in Matrix Multiplication
When performing matrix multiplication, the dimensions of the matrices involved follow specific rules. Let's consider two matrices, matrix P and matrix Q, that are being multiplied to produce a resulting matrix PQ.
If matrix P has dimensions \(m \times n\) and matrix Q has dimensions \(p \times q\), the matrix product PQ is defined only if the number of columns in P (\(n\)) is equal to the number of rows in Q (\(p\)).
If the multiplication is possible (i.e., \(n = p\)), the resulting matrix PQ will have dimensions \(m \times q\). That is, the number of rows in PQ is the number of rows in P, and the number of columns in PQ is the number of columns in Q.
Applying the Rule to Find Matrix B Dimensions
In this question, we are given:
- Matrix A has dimensions \(2 \times 3\).
- The product matrix AB has dimensions \(2 \times 5\).
- We need to find the dimensions of matrix B.
Let the dimensions of matrix B be \(p \times q\).
According to the rules of matrix multiplication:
- For the product AB to be defined, the number of columns in A must equal the number of rows in B. The number of columns in A is 3. The number of rows in B is \(p\). Therefore, \(p = 3\).
- The resulting matrix AB has dimensions given by (number of rows in A) \(\times\) (number of columns in B). The number of rows in A is 2. The number of columns in B is \(q\). The dimensions of AB are given as \(2 \times 5\). Therefore, \(2 \times q = 2 \times 5\), which implies \(q = 5\).
Combining these results, the dimensions of matrix B must be \(p \times q = 3 \times 5\).
Comparing with Given Options
We found that matrix B must have dimensions \(3 \times 5\). Let's look at the provided options:
| Option | Dimensions |
|---|---|
| 1 | \(3 \times 5\) matrix |
| 2 | \(5 \times 3\) matrix |
| 3 | \(3 \times 2\) matrix |
| 4 | \(5 \times 4\) matrix |
Our derived dimensions for matrix B, \(3 \times 5\), match Option 1.
Detailed Explanation of Matrix Multiplication Dimensions
Let's summarize the rule with a general example:
If matrix M has dimensions \(m \times n\) and matrix N has dimensions \(n \times p\), then the product MN is defined and has dimensions \(m \times p\). Notice that the inner dimensions (n and n) must match, and the outer dimensions (m and p) give the dimensions of the result.
In our problem, A is \(2 \times 3\) and AB is \(2 \times 5\). Let B be \(p \times q\).
A\(_{(2 \times 3)}\) \(\times\) B$_{(p \times q)}$ \(=\) AB$_{(2 \times 5)}$
For multiplication to be valid, the number of columns in A must equal the number of rows in B: \(3 = p\).
The resulting matrix AB has dimensions (rows of A) \(\times\) (columns of B): \(2 \times q\). We are given that AB is \(2 \times 5\). So, \(2 \times q = 2 \times 5\), which means \(q = 5\).
Therefore, the dimensions of B are \(p \times q = 3 \times 5\).
Revision Table: Key Concepts in Matrix Dimensions
| Concept | Description | Requirement for Multiplication (A × B) | Resulting Dimension (A × B) |
|---|---|---|---|
| Matrix Dimensions | Number of rows \(\times\) Number of columns | Number of columns in A must equal Number of rows in B | (Number of rows in A) \(\times\) (Number of columns in B) |
| Example: A is \(m \times n\), B is \(p \times q\) | \(n = p\) | \(m \times q\) | |
| Our Case: A is \(2 \times 3\), AB is \(2 \times 5\) | Let B be \(p \times q\) | \(3 = p\) | \(2 \times q = 2 \times 5\), so \(q = 5\) |
Additional Information: Why Dimensions Matter in Matrix Multiplication
Matrix multiplication is not simply multiplying corresponding elements. Each element in the resulting matrix is calculated as the sum of products of elements from a row of the first matrix and a column of the second matrix. This process requires that the 'inner' dimensions match. Specifically, the number of elements in each row of the first matrix must equal the number of elements in each column of the second matrix so that the element-wise multiplication and summation can occur. If the dimensions do not match (\(n \neq p\) in the \(m \times n\) times \(p \times q\) example), the multiplication is undefined. The dimensions of the resulting matrix are determined by the 'outer' dimensions, representing the total number of rows from the first matrix and the total number of columns from the second matrix.
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