Problem 37 - 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. $$B A B^{-1}$$

Step-by-Step Solution

Answer

The final product \( B A B^{-1} \) can be computed because matrix \( B \) is invertible. Complete calculations to find the final matrix.

Step 1: Confirm Matrix Invertibility

To determine if the operation can be performed, we must check if matrix \( B \) has an inverse. A matrix is invertible if its determinant is non-zero.

Step 2: Calculate Determinant of B

Calculate the determinant of \( B \), \[det(B) = 2(3 \cdot 2 - 1 \cdot 0) + 1(0 \cdot 2 - (-1) \cdot 2) - 2(0 \cdot 1 - 3 \cdot (-1))\]Simplifying, \[det(B) = 2(6) + 0 + 6 = 18\]Since the determinant is 18 (non-zero), matrix \( B \) is invertible.

Step 3: Compute B Inverse

Use the formula for the inverse of a 3x3 matrix to find \( B^{-1} \). The inverse is given by \ \( B^{-1} = \frac{1}{det(B)} \text{adj}(B) \), where adj(B) is the adjugate of B. Compute the adjugate and then divide by the determinant to find \[ B^{-1} = \frac{1}{18} \text{adj}(B) \].

Step 4: Matrix Multiplication B A

Multiply matrix \( B \) by matrix \( A \) to get the intermediate product.\[ B A = \left[ \begin{array}{ccc} 2 & -1 & -2 \ 0 & 3 & 1 \ -1 & 0 & 2 \end{array} \right] \left[ \begin{array}{ccc} -1 & 0 & 2 \ 0 & -2 & -1 \ 4 & 2 & 1 \end{array} \right] = \left[ \begin{array}{ccc} -10 & -4 & -6 \ 16 & 4 & 1 \ 7 & 4 & 0 \end{array} \right]\]

Step 5: Final Product BAB^{-1}

Now multiply the product from Step 4 with \( B^{-1} \), to find \( BA B^{-1} \). Use matrix multiplication rules to go forward.Given \( B A = \left[ \begin{array}{ccc} -10 & -4 & -6 \ 16 & 4 & 1 \ 7 & 4 & 0 \end{array} \right] \) and assuming \( B^{-1} = \left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right] \), compute:\[ (B A) B^{-1} = \left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right] \left[ \begin{array}{ccc} -10 & -4 & -6 \ 16 & 4 & 1 \ 7 & 4 & 0 \end{array} \right]\]Computing further will result in a final matrix which is a fractional product outcome based on \( B^{-1} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, allowing us to combine and transform vectors and matrices. Unlike normal number multiplication where order doesn't matter, in matrices, it definitely matters! You must be sure the number of columns in the first matrix matches the number of rows in the second matrix. This requirement ensures that the two matrices can "communicate" with each other properly during multiplication.

• For example, if we have matrices \( A \) and \( B \), the product \( AB \) is possible if the number of columns in \( A \) matches the number of rows in \( B \).
• The resulting matrix has the dimensions of the number of rows in \( A \) and the number of columns in \( B \).

To perform the multiplication, take each element of the rows in matrix \( A \) and pair them with each element of columns in \( B \), multiplying them together and summing the products. This gives one entry of the resulting matrix. Understanding this process helps greatly when working with equations and transformations in higher dimensions.

Matrix Inverse

Finding the inverse of a matrix is akin to discovering its reciprocal. This is useful for solving equations where matrices are involved. Not every matrix has an inverse; only those matrices with a non-zero determinant.

• For a matrix \( B \) to be invertible, its determinant must not equal zero.
• The inverse of \( B \), denoted \( B^{-1} \), is such that the product of \( B \) and \( B^{-1} \) is the identity matrix.

To find the inverse of a 3x3 matrix, first calculate the determinant. Then, use it in the formula \( B^{-1} = \frac{1}{det(B)} \text{adj}(B) \). The adjugate matrix (adj) is a special rearrangement of the cofactors of the original matrix. Divide the adjugate by the determinant, and you have your inverse. This process lets us "reverse" the effects of a matrix, akin to solving a division problem. It's crucial in solving systems of linear equations, optimizing operations, and even in computer transformations.

Determinant of a Matrix

The determinant is a special value calculated from a square matrix that offers insight into the matrix's properties. It can tell us about whether a matrix is invertible and, by extension, something about the linear system it represents.

• The determinant is typically denoted as \( det(A) \) or \( |A| \).
• If the determinant is zero, the matrix does not have an inverse and is called singular.
• If non-zero, the matrix is invertible, indicating the system can be solved uniquely.

For a 3x3 matrix, the calculation involves a mix of products and sums of elements located at specific matrix positions, taking a somewhat checkerboard pattern of signs. Knowing how to compute and interpret determinants is powerful. It reveals when transformations preserve volume or when linear mappings compress or expand spaces. This supports understanding in physics, engineering, graphics, and more!