Problem 54 - 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. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(A B C\)

Step-by-Step Solution

Answer

The product ABC is a 3x3 matrix: \[ \begin{bmatrix} 89 & 57 & 89 \\ -13.5 & -10 & -13.5 \\ 58.25 & 40.5 & 58.25 \end{bmatrix} \]

Step 1: Verify Matrix Dimensions

Matrix multiplication is only possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. Let's check the dimensions. Matrix \( A \) is a \( 3 \times 3 \) matrix. Matrix \( B \) is also \( 3 \times 3 \) and matrix \( C \) is \( 3 \times 3 \). Since all are square matrices with the same number of columns in each preceding matrix as the number of rows in the following matrix, the multiplication \( ABC \) is defined.

Step 2: Multiply Matrices A and B

To find the product \( AB \), multiply each row of matrix \( A \) with each column of matrix \( B \). The resulting matrix will be of size \( 3 \times 3 \).1. First Row: - \( (-2)(0.5) + (0)(-4) + (9)(8) = -1 + 0 + 72 = 71 \) - \( (-2)(3) + (0)(1) + (9)(7) = -6 + 0 + 63 = 57 \) - \( (-2)(0) + (0)(6) + (9)(2) = 0 + 0 + 18 = 18 \)2. Second Row: - \( (1)(0.5) + (8)(-4) + (-3)(8) = 0.5 - 32 - 24 = -55.5 \) - \( (1)(3) + (8)(1) + (-3)(7) = 3 + 8 - 21 = -10 \) - \( (1)(0) + (8)(6) + (-3)(2) = 0 + 48 - 6 = 42 \)3. Third Row: - \( (0.5)(0.5) + (4)(-4) + (5)(8) = 0.25 - 16 + 40 = 24.25 \) - \( (0.5)(3) + (4)(1) + (5)(7) = 1.5 + 4 + 35 = 40.5 \) - \( (0.5)(0) + (4)(6) + (5)(2) = 0 + 24 + 10 = 34 \) The product \( AB \) is: \[\begin{bmatrix}71 & 57 & 18 \-55.5 & -10 & 42 \24.25 & 40.5 & 34\end{bmatrix}\]

Step 3: Multiply the Resulting Matrix with Matrix C

Now, multiply the resulting matrix \( AB \) with \( C \). Again, the resulting dimensions will be \( 3 \times 3 \).1. First Row: - \( (71)(1) + (57)(0) + (18)(1) = 71 + 0 + 18 = 89 \) - \( (71)(0) + (57)(1) + (18)(0) = 0 + 57 + 0 = 57 \) - \( (71)(1) + (57)(0) + (18)(1) = 71 + 0 + 18 = 89 \)2. Second Row: - \( (-55.5)(1) + (-10)(0) + (42)(1) = -55.5 + 0 + 42 = -13.5 \) - \( (-55.5)(0) + (-10)(1) + (42)(0) = 0 - 10 + 0 = -10 \) - \( (-55.5)(1) + (-10)(0) + (42)(1) = -55.5 + 0 + 42 = -13.5 \)3. Third Row: - \( (24.25)(1) + (40.5)(0) + (34)(1) = 24.25 + 0 + 34 = 58.25 \) - \( (24.25)(0) + (40.5)(1) + (34)(0) = 0 + 40.5 + 0 = 40.5 \) - \( (24.25)(1) + (40.5)(0) + (34)(1) = 24.25 + 0 + 34 = 58.25 \)The final product \( ABC \) is: \[\begin{bmatrix}89 & 57 & 89 \-13.5 & -10 & -13.5 \58.25 & 40.5 & 58.25\end{bmatrix}\]

Step 4: Conclusion and Verification

The final matrix product \( ABC \) is calculated as \[\begin{bmatrix}89 & 57 & 89 \-13.5 & -10 & -13.5 \58.25 & 40.5 & 58.25\end{bmatrix}\]. Verify the calculations using a calculator capable of matrix operations to ensure the accuracy of the result.

Key Concepts

Square Matrices

A square matrix is a type of matrix where the number of rows equals the number of columns, forming a symmetrical shape. These matrices are significant because many operations, such as finding determinants and matrix inverses, require square matrices. In the exercise provided, matrices A, B, and C are all examples of square matrices, each having dimensions of 3x3. This symmetry is essential in matrix multiplication, especially when calculating products among multiple matrices as in the problem at hand.
The importance of recognizing a square matrix lies in its properties:

• Symmetry: Number of rows equals the number of columns.
• Determinability: Only square matrices have determinants.
• Inverse Capability: Only invertible if the determinant is non-zero.

In matrix operations, understanding these properties helps in efficient calculations and validations of operations like finding matrix products. Always check for the square format in potential matrices to maximize the use of these features.

Matrix Dimensions

Matrix dimensions refer to the layout described by the row and column configuration, expressed as 'rows x columns'. This description is crucial as it influences how matrices can interact with each other. For example, if you have a 3x2 matrix, it cannot be multiplied with a 2x4 matrix unless you adjust dimensions to meet the matrix multiplication rule: the number of columns in the first matrix must equal the number of rows in the second.
In the original exercise, all matrices (A, B, C) share the same dimensions of 3x3. This consistency in dimensions is what allows the multiplication operation to proceed smoothly. Key points to remember:

• Matching Rule: Only multiply if the number of columns in the first equals the number of rows in the second.
• Resulting Dimension: If a matrix with dimensions m x n is multiplied by an n x p matrix, the product will be an m x p matrix.

This understanding is fundamental for performing matrix operations correctly and efficiently, ensuring the correct setup for calculation sequences.

Matrix Operations

Matrix operations encompass a range of activities including addition, subtraction, and multiplication. Each operation follows specific rules, particularly multiplication which is most nuanced.
Addition and subtraction are straightforward: matrices can be added or subtracted only if they have the same dimensions. Each element is added or subtracted from its corresponding position in the other matrix. Multiplication, however, is more complex. You need to ensure matching dimensions for rows and columns as discussed earlier. This is shown vividly in the step-by-step solution above, where the multiplication of matrices A and B forms a new 3x3 matrix following these precise rules.
Core operations include:

• Addition/Subtraction: Align by dimension and proceed element-wise.
• Multiplication: Align using row-column rules to form new matrices.
• Transpose: Flip a matrix over its diagonal, swapping rows with columns.

Understanding each of these operations, their rules, and outcomes is essential for exploring advanced concepts like linear transformations and system solutions.

Product of Matrices

The product of matrices is a unique operation that combines two or more matrices through multiplication. In the context of matrices A, B, and C, the operation involves sequential multiplication of these square matrices to achieve a final matrix product.
To multiply matrices, use these steps: for every row of the first matrix, compute the dot product with every column of the second matrix. This produces an entry in the i-th row and j-th column of the resultant matrix. The final matrix product will have a dimension that directly depends on the matrices involved. For instance, multiplying a 3x3 matrix by another 3x3 matrix results in a 3x3 product matrix.
Remember:

• Dot Product: Multiply corresponding elements and sum them up to get each position element in the product matrix.
• Sequence Matters: Matrix multiplication is not commutative - meaning AB ≠ BA.
• Dimension Consistency: Follow rules to ensure dimension compatibility.

Applying the rules of multiplication correctly helps in determining valid sequence operations and understanding advanced applications in numerical problems and theoretical explorations.