Problem 26 - Precalculus: Functions and Graphs

Question

Find \(A B\). $$A=\left[\begin{array}{rrr} 1 & 2 & -3 \\ 4 & -5 & 6 \end{array}\right], \quad B=\left[\begin{array}{rrrr} 1 & -1 & 0 & 2 \\ -2 & 3 & 1 & 0 \\ 0 & 4 & 0 & -3 \end{array}\right]$$

Step-by-Step Solution

Answer

The resultant matrix \( AB \) is \( \begin{bmatrix} -3 & -7 & 2 & 11 \\ 14 & 5 & -5 & -10 \end{bmatrix} \).

Step 1: Check Matrix Dimensions

First, confirm that the number of columns in matrix \( A \) matches the number of rows in matrix \( B \). Matrix \( A \) has 3 columns, and matrix \( B \) has 3 rows, so matrix multiplication is possible.

Step 2: Setup Resultant Matrix Dimensions

The resulting matrix \( AB \) will have the same number of rows as matrix \( A \) and the same number of columns as matrix \( B \). Thus, \( AB \) will be a \( 2 \times 4 \) matrix.

Step 3: Calculate Element at First Row, First Column

To find the element at \( (1, 1) \) in \( AB \), multiply elements: \( 1 \times 1 + 2 \times (-2) + (-3) \times 0 = 1 - 4 + 0 = -3 \).

Step 4: Calculate Element at First Row, Second Column

For the element at \( (1, 2) \), compute: \( 1 \times (-1) + 2 \times 3 + (-3) \times 4 = -1 + 6 - 12 = -7 \).

Step 5: Calculate Element at First Row, Third Column

For the element at \( (1, 3) \), compute: \( 1 \times 0 + 2 \times 1 + (-3) \times 0 = 0 + 2 + 0 = 2 \).

Step 6: Calculate Element at First Row, Fourth Column

For the element at \( (1, 4) \), compute: \( 1 \times 2 + 2 \times 0 + (-3) \times (-3) = 2 + 0 + 9 = 11 \).

Step 7: Calculate Element at Second Row, First Column

For the element at \( (2, 1) \), compute: \( 4 \times 1 + (-5) \times (-2) + 6 \times 0 = 4 + 10 + 0 = 14 \).

Step 8: Calculate Element at Second Row, Second Column

For the element at \( (2, 2) \), compute: \( 4 \times (-1) + (-5) \times 3 + 6 \times 4 = -4 - 15 + 24 = 5 \).

Step 9: Calculate Element at Second Row, Third Column

For the element at \( (2, 3) \), compute: \( 4 \times 0 + (-5) \times 1 + 6 \times 0 = 0 - 5 + 0 = -5 \).

Step 10: Calculate Element at Second Row, Fourth Column

For the element at \( (2, 4) \), compute: \( 4 \times 2 + (-5) \times 0 + 6 \times (-3) = 8 + 0 - 18 = -10 \).

Step 11: Write the Resultant Matrix

Combine the calculated elements into the resultant matrix: \( AB = \begin{bmatrix} -3 & -7 & 2 & 11 \ 14 & 5 & -5 & -10 \end{bmatrix} \).

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. It's a fundamental area of mathematics, essential for both theoretical and applied purposes. In particular, matrix multiplication, a key operation in linear algebra, plays a crucial role in numerous applications such as computer graphics, quantum physics, and optimization. Understanding how to multiply matrices not only helps solve systems of linear equations but also aids in the transformation of geometrical figures and the representation of complex networks.

Matrix Dimensions

Matrix dimensions are critical when working with matrices and performing operations such as addition, subtraction, and particularly multiplication. A matrix's dimensions are described in terms of the number of rows and columns it contains. For example, a matrix with 2 rows and 3 columns is a 2x3 matrix. When it comes to matrix multiplication, understanding dimensions is essential to determine if the operation is possible. This requires that the number of columns in the first matrix equals the number of rows in the second matrix. Without matching dimensions, the multiplication cannot be performed, which underlines the importance of checking dimensions before proceeding with the operation. In our example, the matrix A is a 2x3 matrix and matrix B is a 3x4 matrix. The inner dimensions (3 and 3) match, allowing for the multiplication to take place, resulting in a new matrix with dimensions 2x4.

Resultant Matrix

After multiplying two matrices, the product is known as the resultant matrix. The dimensions of the resultant matrix are determined by the outer dimensions of the original matrices involved in the multiplication. So, if you multiply a matrix of dimensions m x n with a matrix of dimensions n x p, the resultant matrix will have dimensions m x p. For our matrices A and B, matrix A is 2x3 and matrix B is 3x4. Accordingly, their product, the resultant matrix, will have the dimensions 2x4, corresponding to the number of rows from matrix A and the number of columns from matrix B. The resultant matrix synthesizes the information from matrices A and B, creating a new matrix reflecting their combined properties.

Element Calculation

Element calculation in matrix multiplication involves computing each entry of the resultant matrix by taking the dot product of the corresponding row from the first matrix and the column from the second matrix.To compute an element located at position (i, j) in the resultant matrix, you follow these steps:

• Take the i-th row from the first matrix.
• Take the j-th column from the second matrix.
• Multiply corresponding elements in the row and column together, producing intermediate products.
• Sum all these intermediate products to get the final entry for position (i, j).

For example, let’s calculate the first element in the first row of the resultant matrix in our exercise:a. From the first matrix A, take the first row: [1, 2, -3]b. From the second matrix B, take the first column: [1, -2, 0]c. Compute the dot product: \( 1 \times 1 + 2 \times (-2) + (-3) \times 0 = 1 - 4 + 0 = -3 \)This methodical process ensures precision in building the entire resultant matrix.