Problem 23 - Precalculus: Functions and Graphs

Question

Find \(A B\). $$A=\left[\begin{array}{rr} 4 & -2 \\ 0 & 3 \\ -7 & 5 \end{array}\right], \quad B=\left[\begin{array}{l} 3 \\ 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix AB is \( \begin{bmatrix} 4 \\ 12 \\ -1 \end{bmatrix} \).

Step 1: Understand the Matrix Dimensions

Matrix \(A\) is a \(3 \times 2\) matrix, meaning it has 3 rows and 2 columns. Matrix \(B\) is a \(2 \times 1\) matrix, meaning it has 2 rows and 1 column. For matrix multiplication to be possible, the number of columns in \(A\) must match the number of rows in \(B\), which they do in this case.

Step 2: Setup for Matrix Multiplication

Since the matrices are compatible for multiplication, we multiply them. The resulting matrix \(AB\) will have dimensions \(3 \times 1\) because we take the number of rows from \(A\) and the number of columns from \(B\).

Step 3: Multiply the First Row of A with B

The first element of \(AB\) is the dot product of the first row of \(A\) and \(B\). Compute: \[ 4 \times 3 + (-2) \times 4 = 12 - 8 = 4 \]

Step 4: Multiply the Second Row of A with B

The second element of \(AB\) is the dot product of the second row of \(A\) and \(B\). Compute: \[ 0 \times 3 + 3 \times 4 = 0 + 12 = 12 \]

Step 5: Multiply the Third Row of A with B

The third element of \(AB\) is the dot product of the third row of \(A\) and \(B\). Compute: \[ (-7) \times 3 + 5 \times 4 = -21 + 20 = -1 \]

Step 6: Write the Resulting Matrix

Combine the results from Steps 3, 4, and 5 to form the resulting matrix: \[ AB = \begin{bmatrix} 4 \ 12 \ -1 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial when dealing with matrix multiplication. The dimensions of a matrix are defined as the number of its rows and columns. For instance, Matrix \(A\) here is a \(3 \times 2\) matrix. This means it has 3 rows and 2 columns. Matrix \(B\) has dimensions \(2 \times 1\). It consists of 2 rows and 1 column.

When we refer to matrix dimensions, it is always in terms of rows by columns:\(m \times n\). Understanding these measurements is essential in determining whether matrices can be multiplied together.

Dot Product

The dot product is a critical operation in matrix multiplication. It involves multiplying corresponding elements of a row and a column, and then summing up the products.

Let's consider matrix \(A\) and matrix \(B\). To find an element in the resulting matrix, take the dot product of a row in \(A\) with the column in \(B\). For instance:

• First row of \(A\) \([4, -2]\) dot product with \(B\) gives \(4 \times 3 + (-2) \times 4 = 12 - 8 = 4\).
• Second row of \(A\) \([0, 3]\) with \(B\) gives \(0 \times 3 + 3 \times 4 = 0 + 12 = 12\).
• Third row of \(A\) \([-7, 5]\) with \(B\) gives \((-7) \times 3 + 5 \times 4 = -21 + 20 = -1\).

This step-by-step dot product process is repeated for each row to fill the resulting matrix.

Matrix Compatibility

Matrix compatibility is crucial to performing multiplication. Two matrices can be multiplied only if the number of columns in the first matrix equals the number of rows in the second matrix.

In this example with matrices \(A\) and \(B\):

• Matrix \(A\) has dimensions \(3 \times 2\).
• Matrix \(B\) has dimensions \(2 \times 1\).

Because the number of columns in \(A\) (2) matches the number of rows in \(B\) (2), these matrices are compatible for multiplication. Without this compatibility, matrix multiplication would be impossible.

Resulting Matrix Dimensions

After confirming matrix compatibility, we derive the resulting matrix dimensions. The result is determined by the number of rows in the first matrix and the number of columns in the second matrix.

Given:

• Matrix \(A\) has 3 rows.
• Matrix \(B\) has 1 column.

Therefore, the resulting matrix after multiplying \(A\) and \(B\) will be a \(3 \times 1\) matrix. It will have three rows (from \(A\)) and one column (from \(B\)), forming:
\[AB = \begin{bmatrix} 4 \ 12 \ -1 \end{bmatrix}\] This result encompasses all the dot products calculated during the solve process.