Problem 17 - Algebra for College Students

Question

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} 2 & -4 \\ 1 & -2 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & -2 \\ -3 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

AB = \(\begin{pmatrix}14 & -28 \\ 7 & -14\end{pmatrix}\); BA = \(\begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}\).

Step 1: Understand the Problem

We are given two matrices, \(A\) and \(B\), and we need to compute their products in both orders: \(AB\) and \(BA\). Matrix multiplication is not commutative, meaning \(AB\) might not equal \(BA\).

Step 2: Compute AB

To compute \(AB\), multiply the rows of matrix \(A\) with the columns of matrix \(B\):\[AB = \left[\begin{array}{cc}2 \cdot 1 + (-4) \cdot (-3) & 2 \cdot (-2) + (-4) \cdot 6\1 \cdot 1 + (-2) \cdot (-3) & 1 \cdot (-2) + (-2) \cdot 6\end{array}\right]\]Calculate each element:- First row, first column: \(2 \cdot 1 + (-4) \cdot (-3) = 2 + 12 = 14\)- First row, second column: \(2 \cdot (-2) + (-4) \cdot 6 = -4 - 24 = -28\)- Second row, first column: \(1 \cdot 1 + (-2) \cdot (-3) = 1 + 6 = 7\)- Second row, second column: \(1 \cdot (-2) + (-2) \cdot 6 = -2 - 12 = -14\)Thus, \[AB = \left[\begin{array}{cc}14 & -28 \ 7 & -14\end{array}\right]\]

Step 3: Compute BA

To compute \(BA\), multiply the rows of matrix \(B\) with the columns of matrix \(A\):\[BA = \left[\begin{array}{cc}1 \cdot 2 + (-2) \cdot 1 & 1 \cdot (-4) + (-2) \cdot (-2)\-3 \cdot 2 + 6 \cdot 1 & -3 \cdot (-4) + 6 \cdot (-2)\end{array}\right]\]Calculate each element:- First row, first column: \(1 \cdot 2 + (-2) \cdot 1 = 2 - 2 = 0\)- First row, second column: \(1 \cdot (-4) + (-2) \cdot (-2) = -4 + 4 = 0\)- Second row, first column: \(-3 \cdot 2 + 6 \cdot 1 = -6 + 6 = 0\)- Second row, second column: \(-3 \cdot (-4) + 6 \cdot (-2) = 12 - 12 = 0\)Thus, \[BA = \left[\begin{array}{cc}0 & 0 \ 0 & 0\end{array}\right]\]

Key Concepts

Non-Commutative Property

Matrix multiplication behaves differently from regular multiplication of numbers. An important aspect of matrix multiplication is the non-commutative property. This means that the order in which you multiply matrices matters. Specifically, if you multiply matrix \(A\) by matrix \(B\), the product \(AB\) is often not the same as \(BA\).
In the example, when calculating \(AB\) and \(BA\) with matrices \(A\) and \(B\) given, we see different resulting matrices.

• \(AB = \left[ \begin{array}{cc} 14 & -28 \ 7 & -14 \end{array} \right]\)
• \(BA = \left[ \begin{array}{cc} 0 & 0 \ 0 & 0 \end{array} \right]\)

Clearly, \(AB eq BA\) which perfectly illustrates the non-commutative property. It is crucial when dealing with matrices to remember that changing the order of multiplication can yield completely different results.

Matrix Product Calculation

Calculating the product of two matrices involves a specific procedure. This process requires multiplying the elements of the rows of the first matrix by the corresponding elements of the columns of the second matrix and then summing these products.
For example, to find the product \(AB\):

• Multiply each element of the first row of \(A\) by each corresponding element of the first column of \(B\), and sum the results for the first element of the product matrix.
• Continue this for the second element of the first row with the second column and repeat for all elements.

This results in computing for \(AB\):

• \(2 \times 1 + (-4) \times (-3) = 14\)
• \(2 \times (-2) + (-4) \times 6 = -28\)
• Follow the same pattern for all elements.

Through this calculation method, you can determine the resulting matrix from two matrices, ensuring accurate computations.

Matrix Operations

Matrix operations are the set of rules and procedures used to perform calculations with matrices, including addition, subtraction, scalar multiplication, and matrix multiplication.

• Addition and Subtraction: Matrices can only be added or subtracted if they have the same dimensions. Perform these operations element-wise.
• Scalar Multiplication: Each element of the matrix is multiplied by the scalar value.
• Matrix Multiplication: This requires a specific method where the number of columns in the first matrix must match the number of rows in the second matrix.

Matrix multiplication can be complex, but following the step-by-step rules ensures the precise calculation of matrix products. Remember, each type of matrix operation follows its distinct rules that must be respected to achieve valid results. Understanding these concepts is critical in fields such as computer graphics, engineering, and mathematics.