Problem 19 - Algebra for College Students

Question

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} -3 & -2 \\ -4 & -1 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ 4 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

AB = \([-14, -7; -12, -1]\), BA = \([-2, -3; -32, -13]\).

Step 1: Computing AB (Matrix Product)

To calculate the matrix product \(AB\), we need to multiply each row of matrix \(A\) with each column of matrix \(B\). The resultant matrix will have the same number of rows as \(A\) and the same number of columns as \(B\).- Multiply the first row of \(A\), \([-3, -2]\), with the first column of \(B\), \([2, 4]\): \((-3 \times 2) + (-2 \times 4) = -6 - 8 = -14\).- Multiply the first row of \(A\), \([-3, -2]\), with the second column of \(B\), \([-1, 5]\): \((-3 \times -1) + (-2 \times 5) = 3 - 10 = -7\).- Multiply the second row of \(A\), \([-4, -1]\), with the first column of \(B\), \([2, 4]\): \((-4 \times 2) + (-1 \times 4) = -8 - 4 = -12\).- Multiply the second row of \(A\), \([-4, -1]\), with the second column of \(B\), \([-1, 5]\): \((-4 \times -1) + (-1 \times 5) = 4 - 5 = -1\).Thus, the product \(AB\) is: \[AB = \begin{bmatrix} -14 & -7 \ -12 & -1 \end{bmatrix}\]

Step 2: Computing BA (Matrix Product)

To calculate the matrix product \(BA\), we need to multiply each row of matrix \(B\) with each column of matrix \(A\). The resultant matrix will have the same number of rows as \(B\) and the same number of columns as \(A\).- Multiply the first row of \(B\), \([2, -1]\), with the first column of \(A\), \([-3, -4]\): \((2 \times -3) + (-1 \times -4) = -6 + 4 = -2\).- Multiply the first row of \(B\), \([2, -1]\), with the second column of \(A\), \([-2, -1]\): \((2 \times -2) + (-1 \times -1) = -4 + 1 = -3\).- Multiply the second row of \(B\), \([4, 5]\), with the first column of \(A\), \([-3, -4]\): \((4 \times -3) + (5 \times -4) = -12 - 20 = -32\).- Multiply the second row of \(B\), \([4, 5]\), with the second column of \(A\), \([-2, -1]\): \((4 \times -2) + (5 \times -1) = -8 - 5 = -13\).Thus, the product \(BA\) is:\[BA = \begin{bmatrix} -2 & -3 \ -32 & -13 \end{bmatrix}\]

Key Concepts

Matrix Product

Matrix multiplication, also known as the matrix product, is an essential operation in linear algebra. It involves multiplying two matrices to produce a new matrix.
Here’s a step-by-step breakdown:

• For two matrices to be multiplied, the number of columns in the first must match the number of rows in the second.
• The result of multiplying an \( m \times n \) matrix by an \( n \times p \) matrix is an \( m \times p \) matrix.
• Each element in the resulting matrix is calculated by taking the dot product of a row from the first matrix and a column from the second matrix.

Understanding this process is crucial as it forms the basis for more advanced matrix operations. When learning matrix multiplication, it’s important to practice with different sets of matrices to become familiar with the pattern of operations.

Matrix Operations

Matrix operations encompass a variety of procedures such as addition, subtraction, and multiplication. These operations allow matrices to be manipulated in ways that serve a wide range of applications in fields like computer science, physics, and economics.
When dealing with matrix multiplication, it's key to remember:

• The **order** of multiplication matters. In general, \( AB \) is not equal to \( BA \).
• Multiplying by the identity matrix doesn't change the original matrix. This is similar to multiplying a number by 1.
• Zero matrices behave differently in multiplication than in addition; they will turn any matrix they are multiplied by into another zero matrix if they are on the correct side of the multiplication.

These are fundamental properties that form the basis of understanding any problem involving matrices.

Algebra Problems

Solving algebra problems that involve matrices can seem complicated, but breaking them down into manageable steps can simplify the process.
Here is how to approach algebra problems involving matrices:

• First, assess the matrices you are given and determine what is being asked. Is it a product, sum, or another operation?
• Apply the rules of matrix operations: Recall the specific rules for matrix addition, subtraction, and multiplication.
• Perform the calculations step by step. Ensuring each step is verified will help prevent mistakes and solidify your understanding.

Practicing algebra problems with matrices frequently can bolster your ability to conduct these operations confidently.