Problem 36 - Precalculus: Mathematics for Calculus

Question

Products Involving Matrices and Inverses The matrices \(A\) and \(B\) are defined as follows. $$ A=\left[\begin{array}{rrr}-1 & 0 & 2 \\\0 & -2 & -1 \\\4 & 2 & 1\end{array}\right] \quad B=\left[\begin{array}{rrr}2 & -1 & -2 \\\0 & 3 & 1 \\\\-1 & 0 & 2\end{array}\right]$$ Use a graphing calculator to carry out the indicated algebraic operations, or explain why they cannot be performed. State the answer using fractions. $$A B^{-1}$$

Step-by-Step Solution

Answer

Matrix \( A B^{-1} \) can be calculated since \( B \) is invertible and has a non-zero determinant.

Step 1: Check Invertibility of Matrix B

Before finding the inverse of matrix \( B \), we must ensure it is invertible. A matrix is invertible if its determinant is not zero. Calculate the determinant of \( B \).

Step 2: Compute the Determinant of Matrix B

The determinant of a 3x3 matrix \( B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \ b_{31} & b_{32} & b_{33} \end{bmatrix} \) is computed as follows:\[ \text{det}(B) = b_{11}(b_{22}b_{33} - b_{23}b_{32}) - b_{12}(b_{21}b_{33} - b_{23}b_{31}) + b_{13}(b_{21}b_{32} - b_{22}b_{31}) \]For our matrix \( B \), calculate:\[ \text{det}(B) = 2(3 \cdot 2 - 1 \cdot 0) - (-1)(0 \cdot 2 - 1 \cdot (-1)) + (-2)(0 \cdot 0 - 3 \cdot (-1)) \]\[ = 2(6) + 1(-1) + 2 \cdot 3 \]\[ = 12 + 1 + 6 = 19 \]

Step 3: Find B Inverse

Since \( \text{det}(B) eq 0 \), matrix \( B \) is invertible. Use a graphing calculator to find \( B^{-1} \). The inverse will be a 3x3 matrix with fractions.

Step 4: Calculate AB^{-1}

Once \( B^{-1} \) is found using the graphing calculator, calculate the matrix product \( A B^{-1} \). Perform the matrix multiplication by taking the dot product of the rows of \( A \) with the columns of \( B^{-1} \), using the values generated by the calculator.

Step 5: Express the Result as Fractions

Ensure that the entries in the resulting matrix \( A B^{-1} \) are expressed as fractions, as required.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. When it comes to matrix calculations, especially for a 3x3 matrix, the determinant plays a crucial role. Understanding this concept is key to determining whether a matrix is invertible.
To find the determinant of a 3x3 matrix, use the following formula:

• Multiply the elements in sequence starting from the first element of the first row, followed by specific cross-products from lower-submatrices.
• For example, in matrix B from the problem, the calculation is as follows: \[ ext{det}(B) = 2(3 \cdot 2 - 1 \cdot 0) - (-1)(0 \cdot 2 - 1 \cdot (-1)) + (-2)(0 \cdot 0 - 3 \cdot (-1)) \ = 2(6) + 1(-1) + 2 \cdot 3 = 19 \ \]
• If the determinant is not zero, the matrix is invertible.

This brief calculation determines the matrix's invertibility, crucial for further operations like finding the inverse.

Matrix Multiplication

Matrix multiplication is a mathematical operation where two matrices are combined to yield another matrix. This operation is more intricate than simple arithmetic multiplication.
Matrix multiplication involves the dot product of rows from the first matrix with columns from the second matrix. Here’s how you can perform matrix multiplication step by step:

• Ensure both matrices are compatible for multiplication. For matrices of sizes m×n and n×p, the resulting matrix will be of size m×p.
• For each element in the resulting matrix, compute the sum of products of corresponding elements from rows of the first matrix and columns of the second matrix.
• In this exercise, after obtaining the inverse of B, you calculate \( AB^{-1} \).

Remember, in some cases, using a graphing calculator for the dot products can ease the process, especially with larger matrices involving fractions. The final matrix should be expressed with each element converted into fractions for precision.

3x3 Matrix

A 3x3 matrix is simply a matrix with three rows and three columns. It's one of the fundamental matrix structures used in linear algebra and is commonly encountered in problems involving transformations.
Key properties and operations associated with a 3x3 matrix include:

• The ability to calculate determinants, which is essential for identifying inversibility.
• Performing operations such as matrix addition, subtraction, and multiplication, where understanding positional indexes is crucial.
• Inversion, only possible if the matrix has a non-zero determinant.

3x3 matrices are often used to solve systems of equations, handle transformations in geometry, and assist in various calculations in engineering and computer science. Mastery of this matrix size is foundational for more complex matrix operations and applications.