Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(C A\)

Step-by-Step Solution

Answer

The result of multiplying C by A is \( \begin{bmatrix} -1.5 & 4 & 14 \\ 1 & 8 & -3 \\ -1.5 & 4 & 14 \end{bmatrix} \).

Step 1: Determine if Matrix Multiplication is Possible

Matrix multiplication is possible only when the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix \(C\) is a \(3 \times 3\) matrix and matrix \(A\) is also \(3 \times 3\). Since both matrices have dimensions \(3 \times 3\), matrix multiplication is possible.

Step 2: Set Up the Matrix Multiplication Operation

We need to calculate \(C \times A\). Write out both matrices in a neat, aligned manner:\[ C = \begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} -2 & 0 & 9 \ 1 & 8 & -3 \ 0.5 & 4 & 5 \end{bmatrix} \]

Step 3: Perform Matrix Multiplication

Multiply each row element of matrix \(C\) by the corresponding column element of matrix \(A\) and add the products to get each element of the resulting matrix.- Element \((1,1)\) of \(C \times A\): \((1 \times -2) + (0 \times 1) + (1 \times 0.5) = -2 + 0 + 0.5 = -1.5\)- Element \((1,2)\) of \(C \times A\): \((1 \times 0) + (0 \times 8) + (1 \times 4) = 0 + 0 + 4 = 4\)- Element \((1,3)\) of \(C \times A\): \((1 \times 9) + (0 \times -3) + (1 \times 5) = 9 + 0 + 5 = 14\)- Element \((2,1)\) of \(C \times A\): \((0 \times -2) + (1 \times 1) + (0 \times 0.5) = 0 + 1 + 0 = 1\)- Element \((2,2)\) of \(C \times A\): \((0 \times 0) + (1 \times 8) + (0 \times 4) = 0 + 8 + 0 = 8\)- Element \((2,3)\) of \(C \times A\): \((0 \times 9) + (1 \times -3) + (0 \times 5) = 0 - 3 + 0 = -3\)- Element \((3,1)\) of \(C \times A\): \((1 \times -2) + (0 \times 1) + (1 \times 0.5) = -2 + 0 + 0.5 = -1.5\)- Element \((3,2)\) of \(C \times A\): \((1 \times 0) + (0 \times 8) + (1 \times 4) = 0 + 0 + 4 = 4\)- Element \((3,3)\) of \(C \times A\): \((1 \times 9) + (0 \times -3) + (1 \times 5) = 9 + 0 + 5 = 14\)

Step 4: Write the Resulting Matrix

Combine all calculated elements to form the resulting matrix:\[ C \times A = \begin{bmatrix} -1.5 & 4 & 14 \ 1 & 8 & -3 \ -1.5 & 4 & 14 \end{bmatrix} \]

Key Concepts

Matrix Operations

Matrix operations involve various procedures that you can perform on matrices. The most common operations include addition, subtraction, and multiplication.
In matrix multiplication, you take two matrices and produce another matrix. It isn't as straightforward as adding or subtracting, and the operation is defined only under certain conditions.

For multiplication to happen, the number of columns in the first matrix must equal the number of rows in the second matrix. This condition is vital because each element in the product matrix is derived from summing a series of products between elements from corresponding rows and columns.
For example, when multiplying matrices \(C\) and \(A\) in the exercise, each element in the resultant matrix is derived by multiplying corresponding elements from \(C\)'s rows with \(A\)'s columns and adding the results. Each computed value forms one cell in the resulting matrix. If each matrix holds compatible dimensions, multiplication is possible.

Matrices

Matrices are an essential concept in mathematics, particularly in linear algebra. They are essentially arrays of numbers and are a powerful tool to handle simultaneous equations or other similar repetitive calculations.
Matrices come in various dimensions, described by the number of rows and columns they contain. For example, a \(3 \times 3\) matrix has three rows and three columns. These arrays can represent not just numbers but also complex systems and transformations in various fields from physics to computer science.

As seen in our exercise, the matrices \(A\) and \(C\) are both \(3 \times 3\) matrices. Such square matrices often appear in problems because they have straightforward multiplication properties and many useful characteristics, like having determinant values, which help in solving systems of equations.

Linear Algebra

Linear algebra is a branch of mathematics focusing on linear systems of equations and their representations through matrices and vector spaces. It's a fundamental part of many scientific fields.
Linear algebra covers a broad range of topics including vector spaces, linear transformations, and matrices, serving as the foundational language for describing linear relationships and transformations.

In the exercise, matrix multiplication is a crucial part of exploring linear transformations. By using matrices \(A\) and \(C\), the multiplication operation transforms the information encoded in matrices, thereby producing a new matrix reflecting the composition of transformations.

• It's through linear algebra that experts solve complex real-world problems like network flows, optimizations, 3D rotations in computer graphics, and more.
• This subject's principles are foundational to advanced subjects such as machine learning, statistics, and more.