Problem 32 - Algebra 2

Question

Use \(A=\left[\begin{array}{cc}{1} & {-2} \\ {4} & {3}\end{array}\right], B=\left[\begin{array}{cc}{-5} & {2} \\ {4} & {3}\end{array}\right], C=\left[\begin{array}{cc}{5} & {1} \\ {2} & {-4}\end{array}\right]\) and scalar \(c=3\) to determine whether the following equations are true for the given matrices. \((A B) C=(C B) A\)

Step-by-Step Solution

Answer

The equation \((AB)C = (CB)A\) is false.

Step 1: Calculate AB

First, let's calculate the product of matrices \( A \) and \( B \). The product of two matrices \( A \) and \( B \) is given by the expression \([A]_{ij} = \sum_{k} A_{ik} B_{kj}\). Use this formula:\[AB = \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix} \cdot \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix}\]Calculate each element:- First row, first column: \((1)(-5) + (-2)(4) = -5 - 8 = -13 \).- First row, second column: \((1)(2) + (-2)(3) = 2 - 6 = -4\).- Second row, first column: \((4)(-5) + (3)(4) = -20 + 12 = -8 \).- Second row, second column: \((4)(2) + (3)(3) = 8 + 9 = 17 \).So, \( AB = \begin{bmatrix} -13 & -4 \ -8 & 17 \end{bmatrix} \).

Step 2: Calculate (AB)C

Now, multiply the result from Step 1 by the matrix \( C \):\[(AB)C = \begin{bmatrix} -13 & -4 \ -8 & 17 \end{bmatrix} \cdot \begin{bmatrix} 5 & 1 \ 2 & -4 \end{bmatrix}\]Calculate each element of the product:- First row, first column: \((-13)(5) + (-4)(2) = -65 - 8 = -73\).- First row, second column: \((-13)(1) + (-4)(-4) = -13 + 16 = 3\).- Second row, first column: \((-8)(5) + (17)(2) = -40 + 34 = -6\).- Second row, second column: \((-8)(1) + (17)(-4) = -8 - 68 = -76\).So, \((AB)C = \begin{bmatrix} -73 & 3 \ -6 & -76 \end{bmatrix} \).

Step 3: Calculate CB

Next, calculate the product of matrices \( C \) and \( B \):\[CB = \begin{bmatrix} 5 & 1 \ 2 & -4 \end{bmatrix} \cdot \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix}\]Calculate each element:- First row, first column: \((5)(-5) + (1)(4) = -25 + 4 = -21\).- First row, second column: \((5)(2) + (1)(3) = 10 + 3 = 13\).- Second row, first column: \((2)(-5) + (-4)(4) = -10 - 16 = -26\).- Second row, second column: \((2)(2) + (-4)(3) = 4 - 12 = -8\).So, \( CB = \begin{bmatrix} -21 & 13 \ -26 & -8 \end{bmatrix} \).

Step 4: Calculate (CB)A

Now, multiply the result from Step 3 by the matrix \( A \):\[(CB)A = \begin{bmatrix} -21 & 13 \ -26 & -8 \end{bmatrix} \cdot \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix}\]Calculate each element:- First row, first column: \((-21)(1) + (13)(4) = -21 + 52 = 31\).- First row, second column: \((-21)(-2) + (13)(3) = 42 + 39 = 81\).- Second row, first column: \((-26)(1) + (-8)(4) = -26 - 32 = -58\).- Second row, second column: \((-26)(-2) + (-8)(3) = 52 - 24 = 28\).So, \((CB)A = \begin{bmatrix} 31 & 81 \ -58 & 28 \end{bmatrix} \).

Step 5: Compare (AB)C and (CB)A

Finally, compare the matrices obtained in Step 2 and Step 4:- \((AB)C = \begin{bmatrix} -73 & 3 \ -6 & -76 \end{bmatrix}\)- \((CB)A = \begin{bmatrix} 31 & 81 \ -58 & 28 \end{bmatrix}\)Since these matrices are not equal, the equation \((AB)C = (CB)A \) does not hold.

Key Concepts

Matrix Algebra

Matrix algebra is a fundamental area in mathematics that allows us to perform operations on matrices, just like we do with numbers. It extends concepts of arithmetic operations like addition, subtraction, and multiplication to a higher dimension: matrices! In matrix algebra, we work with rows and columns of numbers.

Here are the basic operations in matrix algebra:

• Addition and Subtraction: You can add or subtract matrices of the same size by performing the operation on the corresponding elements.
• Scalar Multiplication: You multiply each element of a matrix by a number (known as a scalar).
• Matrix Multiplication: This is a bit trickier. You multiply rows of one matrix by the columns of another. The number of columns in the first matrix must match the number of rows in the second matrix.

Matrix multiplication is not commutative, meaning that the order of multiplication matters!
This concept is essential in understanding our exercise because it dictates how we compute products like \((AB)C\) or \((CB)A\). It's these calculations that tell us more about matrix relationships.

Associative Property

The associative property is a key principle in algebra, including matrix algebra. It explains that when you work with three or more matrices, the way you group them does not change the final product.

For matrices, this property can be expressed as:

• \((AB)C = A(BC)\)

This property only concerns the grouping of matrices, not their order.
But, as we've seen in the exercise, even though

• \((AB)C\)
• \((CB)A\)

seems to suggest this associative property, we are actually changing the order of operations in our exercise, not just the grouping. Therefore the associative property wasn't violated; it was never applicable in the first place! Matrices need to hold equivalent size for this to even consider. Disproving that equality shows how misapplying the property can lead to false conclusions.

Matrix Equation

A matrix equation like \((AB)C = (CB)A\) represents two potential outcomes of matrix operations.
Understanding how to properly execute each multiplication step is crucial to solving such equations. Each side of the equation represents a different method of structuring and solving a problem using known quantities (matrices in this case).

Let's break it down:

• Calculate \((AB)C\): First multiply \(A\) by \(B\), and then multiply the result by \(C\). Each calculation is done by following the matrix multiplication rules.
• Calculate \((CB)A\): Similarly, you start by multiplying \(C\) by \(B\), then multiply the resulting matrix with \(A\).
• Compare: Once both calculations are performed, compare the results. Only if both results are the same can we say the matrices are equal under the given equation.

In this exercise, the results differed, meaning that the original equation \((AB)C = (CB)A\) held false.
This illustrates an essential learning point that the sequence in which we perform tasks truly matters and must be adhered to, especially in matrix equations.