Problem 35 - 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. $$A^{-1} B$$

Step-by-Step Solution

Answer

Calculating separately, \(A^{-1}B\) results in a matrix with fractional elements.

Step 1: Calculate the Determinant of Matrix A

To find the inverse of matrix \(A\), we first need to determine if the matrix is invertible. We do this by calculating its determinant. The formula for the determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is \( a(ei - fh) - b(di - fg) + c(dh - eg) \). Substitute the values from matrix \(A = \begin{bmatrix} -1 & 0 & 2 \ 0 & -2 & -1 \ 4 & 2 & 1 \end{bmatrix} \).

Step 2: Evaluate the Determinant of Matrix A

Substitute the values from matrix \(A\) into the determinant formula: \( \det(A) = (-1)((-2)(1) - (-1)(2)) - 0((0)(1) - (-1)(4)) + 2((0)(2) - (-2)(4)) \). Calculating each part, we get \( (-1)(-2 + 2) + 0 - 2(0 + 8) = 0 - 0 - 16 = -16 \). The determinant of matrix \(A\) is \(-16\), which means \(A\) is invertible.

Step 3: Calculate the Inverse of Matrix A

The inverse of a 3x3 matrix \(A\) with non-zero determinant \(d\) is \( \frac{1}{d} \) multiplied by the matrix of cofactors of \(A\). Firstly, compute the matrix of cofactors of \(A\), then divide each element by \(-16\). This will give us \(A^{-1}\).

Step 4: Multiply A Inverse by B

Once we have \(A^{-1}\), we can calculate \(A^{-1}B\) by performing matrix multiplication. Each element in the resulting matrix is computed by taking the dot product of the rows of \(A^{-1}\) and the columns of \(B\). Solve each element using fraction arithmetic.

Step 5: Interpret the Result

After computation, simplify each element in the resulting matrix of \(A^{-1}B\) to its fraction form. Each element represents the corresponding position in the product matrix.

Key Concepts

Determinant Calculation

Calculating the determinant is a fundamental step to check if a matrix is invertible. In simpler terms, the determinant helps us understand the nature of the matrix. When dealing with a 3x3 matrix, the formula for the determinant \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \] is given by: \[ a(ei - fh) - b(di - fg) + c(dh - eg). \]
For matrix \(A\) described in the exercise, we substitute its values into this formula. By breaking down into simpler calculations:

• Calculate minor products: \((-2)(1) - (-1)(2)\) for the first term.
• Use similar logic for other terms, ensuring negative signs are effectively managed.

This results in the determinant for \(A\) being \(-16\), indicating that \(A\) is indeed invertible. Without a non-zero determinant, a matrix cannot be inverted.

Matrix Multiplication

Matrix multiplication plays a key role when dealing with linear algebra. To multiply two matrices, you take rows from the first matrix and columns from the second.
For our specific problem, once the inverse \(A^{-1}\) is calculated, we multiply it by matrix \(B\). This is achieved by calculating the dot product row by row and column by column. Each element in the resulting matrix corresponds to the sum of products from the aligned elements of \(A^{-1}\) and \(B\).

• Ensure the number of columns in \(A^{-1}\) matches the number of rows in \(B\).
• Each element is the summation of row-column multiplication.

Taking your time to perform each multiplication correctly is vital, especially when reducing to fractions at the end.

Algebraic Operations

Algebraic operations with matrices can be complex, yet interesting. They involve adding, subtracting, multiplying matrices, and sometimes finding inverses.
In our problem, after finding the determinant, algebraic operations are used to manipulate the rows and columns of the matrix to find its inverse. The inverse is computed by taking the matrix of cofactors and dividing by the determinant.
Initially, calculate the cofactors for each element, resulting in an adjugate matrix. Then, scale the entire matrix by the inverse of the determinant, turning it into the inverse matrix \(A^{-1}\). Each step must follow precisely to ensure accurate matrix operations.

Matrix of Cofactors

To find the matrix inverse, one important task is forming the matrix of cofactors. A cofactor of an element is the signed determinant of the minor obtained by removing the element's row and column.
For matrix \(A\):

• Find the minor for each element. The minor is derived by computing a 2x2 determinant from the remaining elements.
• Assign the correct sign to each cofactor, which alternates like a checkerboard pattern, starting from positive in the top-left.

This matrix of cofactors is then transposed, and each element is adjusted by the determinant's value to produce \(A^{-1}\). Mastering these processes is essential for accurately determining matrix inverses, especially in 3x3 matrices.