Question

Given the following four matrices, test whether any one of them is the inverse of another: \\[ D=\left[\begin{array}{rr} 1 & 12 \\ 0 & 3 \end{array}\right] \quad E=\left[\begin{array}{rr} 1 & 1 \\ 6 & 8 \end{array}\right] \quad F=\left[\begin{array}{rr} 1 & -4 \\ 0 & \frac{1}{3} \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & -\frac{1}{2} \\ -3 & \frac{1}{2} \end{array}\right] \\]

Short answer

Matrices \( D \) and \( F \) are inverses, and matrices \( E \) and \( C \) are inverses of each other.

Step-by-step solution

Understanding Matrix Inverses

A matrix \( A \) will have an inverse \( A^{-1} \) if and only if \( A \times A^{-1} = I \), where \( I \) is the identity matrix for the same order. For a \( 2 \times 2 \) matrix, the identity matrix is \( \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] \). Our task is to check if any pair of matrices among \( D, E, F, C \) are inverses of each other.

Check Pair (D and F)

Compute \( D \times F \) and \( F \times D \) and verify if either results in the identity matrix. \[ D \times F = \left[\begin{array}{cc} 1 & 12 \ 0 & 3 \end{array}\right] \times \left[\begin{array}{cc} 1 & -4 \ 0 & \frac{1}{3} \end{array}\right] = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] \] This shows \( D \times F = I \). \[ F \times D = \left[\begin{array}{cc} 1 & -4 \ 0 & \frac{1}{3} \end{array}\right] \times \left[\begin{array}{cc} 1 & 12 \ 0 & 3 \end{array}\right] = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] \] Therefore, \( F \times D = I \), confirming \( F \) and \( D \) are inverses.

Check Other Pairs

Although we've found that \( D \) and \( F \) are inverses, we should verify other pairs as well for completeness. Compute \( E \times C \) and \( C \times E \) to see if any result in the identity matrix. \[ E \times C = \left[\begin{array}{cc} 1 & 1 \ 6 & 8 \end{array}\right] \times \left[\begin{array}{cc} 4 & -\frac{1}{2} \ -3 & \frac{1}{2} \end{array}\right] = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] \] \( E \times C = I \). Similarly, \( C \times E = I \) as you will find upon computation. Thus, \( C \) and \( E \) are also inverses.

Key concepts

Matrix Multiplication

Matrix multiplication is a crucial operation in linear algebra, especially when dealing with systems of equations or transformations. It involves combining two matrices to produce a third matrix. Here’s how it works for two matrices, say matrix A and matrix B:

• The number of columns in the first matrix, A, must be equal to the number of rows in the second matrix, B, for the multiplication to be valid.
• Each element of the resulting matrix is computed by taking the dot product of the corresponding row of the first matrix and the column of the second matrix.

When multiplying two 2x2 matrices, you calculate the elements of the resulting matrix by summing the products of the corresponding entries from rows and columns:

• The element in the first row, first column of the result comes from the sum of the first row of A and the first column of B.
• Continue this for all combinations of rows of A and columns of B for the whole resulting matrix.

Matrix multiplication is not commutative; meaning, in general, \( A \times B eq B \times A \). However, it can be associative and distributive, and understanding these properties is essential to solving problems in linear algebra.

Identity Matrix

The identity matrix plays a pivotal role when working with matrix operations, much like the number 1 plays in multiplication with regular numbers. Here are some key points about the identity matrix:

• It's a square matrix, meaning it has the same number of rows and columns.
• All elements on the diagonal from the top left to the bottom right are 1’s, and all other elements are zeros.
• In a 2x2 identity matrix, this looks like \( \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \).
• When you multiply any matrix by the identity matrix, it remains unchanged: \( A \times I = A \).

This property means that the identity matrix is the multiplicative identity in the matrix world, confirming the inverse relationship. If two matrices result in the identity matrix when multiplied in both orders, then these matrices are considered inverses of each other.

2x2 Matrix Inverse Calculation

Finding the inverse of a 2x2 matrix is a fundamental task in algebra. An inverse for a matrix A is another matrix, denoted \( A^{-1} \), such that \( A \times A^{-1} = I \). Here’s how you calculate the inverse of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \):

• First, compute the determinant: \( det(A) = ad - bc \).
• If the determinant is not zero, the matrix is invertible. Otherwise, it is not.
• Assuming the determinant is non-zero, the inverse is computed as:\[A^{-1} = \frac{1}{ad-bc} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right]\]This formula swaps the positions of a and d, and changes the signs of b and c.

The determinant acts as a checking mechanism—if it’s zero, finding an inverse isn’t possible. Therefore, always check the determinant before attempting to calculate an inverse matrix.