Problem 30 - Algebra 2

Question

Determine whether each pair of matrices are inverses of each other. $$ C=\left[\begin{array}{rr}{1} & {5} \\ {1} & {-2}\end{array}\right], D=\left[\begin{array}{rr}{\frac{2}{7}} & {\frac{5}{7}} \\ {\frac{1}{7}} & {-\frac{1}{7}}\end{array}\right] $$

Step-by-Step Solution

Answer

Yes, matrices C and D are inverses of each other.

Step 1: Define Matrix Multiplication

The matrices \( C \) and \( D \) are given, and we need to verify if they are inverses. Two matrices are inverses of each other if their product is the identity matrix. For a 2x2 matrix, the identity matrix is \( I = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] \). Thus, we multiply \( C \) and \( D \) and check if the result is the identity matrix.

Step 2: Multiply Matrices C and D

To find the product of matrices \( C \) and \( D \), we compute each element of the resulting matrix using the formula for matrix multiplication:\[(C \cdot D)_{ij} = \sum_k C_{ik} \cdot D_{kj}\]Calculating the elements:- First row, first column: \((1)(\frac{2}{7}) + (5)(\frac{1}{7}) = \frac{2}{7} + \frac{5}{7} = 1\)- First row, second column: \((1)(\frac{5}{7}) + (5)(-\frac{1}{7}) = \frac{5}{7} - \frac{5}{7} = 0\)- Second row, first column: \((1)(\frac{2}{7}) + (-2)(\frac{1}{7}) = \frac{2}{7} - \frac{2}{7} = 0\)- Second row, second column: \((1)(\frac{5}{7}) + (-2)(-\frac{1}{7}) = \frac{5}{7} + \frac{2}{7} = 1\)The result is the matrix \( \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] \), which is the identity matrix.

Step 3: Conclusion

Since the product of matrices \( C \) and \( D \) is the identity matrix, they are indeed inverses of each other. This means \( C \cdot D = I \) and it confirms the given matrices are inverses.

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra where two matrices are multiplied together to produce a third matrix. This operation is not like regular multiplication because in matrix multiplication, the order matters due to the arrangement of rows and columns. To multiply a matrix row by a column, you calculate the sum of the products of the elements. For example, if you have two 2x2 matrices, like matrix \( C \) and \( D \):

• The entry at the first row and first column of the result is from multiplying and summing the first row of \( C \) with the first column of \( D \).
• Similarly, you follow this method for the rest of the entries, ensuring that for each position, you sum the products of corresponding entries in the row from \( C \) and the column from \( D \).

Ensuring you multiply matrices correctly is essential to finding key results like the identity matrix, whether for solving linear equations or verifying inverse matrices.

Identity Matrix

The identity matrix acts like the number 1 in the world of matrices. It's a square matrix with ones on the diagonal and zeros elsewhere. In a 2x2 identity matrix, it looks like this:\[I = \begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]One of the main properties of the identity matrix is that when any matrix is multiplied by the identity matrix, the original matrix itself is unchanged. This is true whether the identity matrix is on the left or right in the multiplication. In the context of matrix inverses, if two matrices multiply together to form the identity matrix, they are inverses of one another. When checking for inverse matrices, we seek to establish that this multiplication equals the identity matrix.

2x2 Matrices

2x2 matrices are among the simplest forms of matrices and are commonly used in many basic linear algebra concepts. A 2x2 matrix is simply a grid containing 2 rows and 2 columns. When analyzing a 2x2 matrix, there are a few critical operations to be aware of, including determinant calculation and finding inverses.

• For a 2x2 matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \], its determinant is calculated as \( ad - bc \).
• The inverse of a non-singular 2x2 matrix \( A \), if it exists, is given by \[ \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \div (ad - bc) \]. This is the key formula used to find these inverses.

Understanding these computations aids in the verification of whether two given 2x2 matrices are inverses, which involves ensuring that when they are multiplied, they yield the identity matrix. This ability to verify inverses is essential in studying systems of linear equations and transformations.