Problem 12 - College Algebra

Question

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$A=\left[\begin{array}{rrr}3 & 8 & 2 \\ 1 & 1 & 1 \\ 5 & 6 & 12\end{array}\right], B=\frac{1}{36}\left[\begin{array}{rrr}-6 & 84 & -6 \\ 7 & -26 & 1 \\ -1 & -22 & 5\end{array}\right]$$

Step-by-Step Solution

Answer

Matrix A is the inverse of Matrix B because their product is the identity matrix.

Step 1: Understand the Matrix Inversion Condition

Two matrices, \(A\) and \(B\), are inverses of each other if their product, \(AB\), results in the identity matrix \(I\). For a 3x3 matrix, the identity matrix is \(I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\).

Step 2: Calculate the Product AB

Compute the product of matrices \(A\) and \(B\) by performing matrix multiplication. Each element in the resulting matrix is obtained by taking the dot product of the rows of \(A\) and the columns of \(B\).

Step 3: Matrix Multiplication Details

- Calculate the first row of \(AB\): \[(3)(-\frac{1}{6}) + (8)(7) + (2)(-\frac{1}{6})=1\] \[(3)(\frac{84}{36}) + (8)(-\frac{26}{36}) + (2)(-\frac{22}{36})=0\] \[(3)(-\frac{6}{36}) + (8)(\frac{1}{36}) + (2)(\frac{5}{36})=0\]- Calculate the second row of \(AB\): \[(1)(-\frac{1}{6}) + (1)(7) + (1)(-\frac{1}{6})=0\] \[(1)(\frac{84}{36}) + (1)(-\frac{26}{36}) + (1)(-\frac{22}{36})=1\] \[(1)(-\frac{6}{36}) + (1)(\frac{1}{36}) + (1)(\frac{5}{36})=0\]- Calculate the third row of \(AB\): \[(5)(-\frac{1}{6}) + (6)(7) + (12)(-\frac{1}{6})=0\] \[(5)(\frac{84}{36}) + (6)(-\frac{26}{36}) + (12)(-\frac{22}{36})=0\] \[(5)(-\frac{6}{36}) + (6)(\frac{1}{36}) + (12)(\frac{5}{36})=1\]The calculated product of \(AB\) is the identity matrix, confirming \(A\) is the inverse of \(B\).

Step 4: Verify Identity Matrix Result

Confirm the product of \(AB\) yields:\[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]This verifies that \(A\) and \(B\) are indeed inverses.

Key Concepts

Identity Matrix

The identity matrix is a fundamental concept in linear algebra. It serves as a neutral or "do-nothing" element in matrix multiplication. In simpler terms, when you multiply any matrix by an identity matrix, the original matrix remains unchanged. This is much like multiplying a number by the number 1 in arithmetic.

For a 3x3 matrix, the identity matrix is:\[I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]

• The main diagonal elements of the identity matrix are all 1s.
• All other elements are 0s.

It's important in operations where you need to check or verify if two matrices are inverses of each other. If multiplying two matrices results in an identity matrix, they are inverses.

Matrix Multiplication

Matrix multiplication is different from the multiplication of numbers. You can't just multiply corresponding elements of matrices directly. Instead, multiply rows by columns, creating new elements. This action is often thought of as taking the dot product between rows of the first matrix and columns of the second matrix.

For a clearer understanding:

• Each element in the resulting matrix is obtained from the sum of the products of the corresponding elements.
• The number of columns in the first matrix must match the number of rows in the second.
• The resulting matrix's dimensions are determined by the number of rows in the first and columns in the second.

This method helps in various applications, such as transforming data or verifying matrix inverses.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. Mathematicians and data scientists frequently use them due to their complexity and detailed nature. They can represent various transformations and solve systems of equations.

Characteristics of a 3x3 matrix include:

• They have a defined determinant, which is crucial for many calculations, including finding inverses.
• They often represent physical systems in 3D space or transformations such as rotations and scaling.

Understanding 3x3 matrices lays the groundwork for more advanced matrix computations, including finding inverses or determinants.

Inverse Matrix Verification

Inverse matrix verification is a process to ensure that two matrices truly are inverses of each other. To prove this, when you multiply one matrix by its supposed inverse, the product should be the identity matrix.

Steps for verification:

• Start with two matrices, say, \(A\) and \(B\).
• Perform matrix multiplication on \(A\) and \(B\).
• If the resulting matrix is the identity matrix (in the 3x3 scenario), then \(A\) and \(B\) are confirmed as inverses.

This process requires accuracy in multiplication to avoid errors. It's a crucial verification technique that confirms relationships between matrices.