Problem 10 - Algebra 2

Question

Determine whether each pair of matrices are inverses of each other. $$ R=\left[\begin{array}{ll}{2} & {2} \\ {3} & {4}\end{array}\right], S=\left[\begin{array}{rr}{2} & {-1} \\ {-\frac{3}{2}} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

R and S are inverses of each other.

Step 1: Verify Matrix Multiplication Validity

To determine if two matrices are inverses, their product must be the identity matrix. Both matrices \( R \) and \( S \) are 2x2 matrices, making them ready for multiplication.

Step 2: Define the Identity Matrix

The identity matrix for 2x2 matrices is \( I = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] \). We will compare the product of \( R \) and \( S \) to this matrix.

Step 3: Multiply Matrices R and S

Calculate the product \( R \times S \):\[R \times S = \begin{bmatrix} 2 & 2 \ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 2 & -1 \ -\frac{3}{2} & 1 \end{bmatrix}\]Calculate each element:\(a_{11} = (2 \times 2) + (2 \times (-\frac{3}{2})) = 4 - 3 = 1\)\(a_{12} = (2 \times -1) + (2 \times 1) = -2 + 2 = 0\)\(a_{21} = (3 \times 2) + (4 \times (-\frac{3}{2})) = 6 - 6 = 0\)\(a_{22} = (3 \times -1) + (4 \times 1) = -3 + 4 = 1\)Thus, \( R \times S = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which is the identity matrix.

Step 4: Multiply Matrices S and R

To ensure they are truly inverses, also calculate \( S \times R \):\[S \times R = \begin{bmatrix} 2 & -1 \ -\frac{3}{2} & 1 \end{bmatrix} \times \begin{bmatrix} 2 & 2 \ 3 & 4 \end{bmatrix}\]Calculate each element:\(a_{11} = (2 \times 2) + (-1 \times 3) = 4 - 3 = 1\)\(a_{12} = (2 \times 2) + (-1 \times 4) = 4 - 4 = 0\)\(a_{21} = (-\frac{3}{2} \times 2) + (1 \times 3) = -3 + 3 = 0\)\(a_{22} = (-\frac{3}{2} \times 2) + (1 \times 4) = -3 + 4 = 1\)Thus, \( S \times R = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which again is the identity matrix.

Step 5: Conclusion

Since both products \( R \times S \) and \( S \times R \) equal the identity matrix, the matrices \( R \) and \( S \) are inverses of each other.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing another matrix.
To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.
For two 2x2 matrices, this condition is naturally satisfied.

• Multiply each element from a row of the first matrix by the corresponding element of a column in the second matrix.
• Sum these products to get an element of the resulting matrix.

This process is applied to get each element of the resulting matrix, effectively combining the matrices' rows and columns.
In the exercise, matrices \( R \) and \( S \) are multiplied to check if their product equals the identity matrix, an essential step to verify if they are inverses.

Identity Matrix

An identity matrix plays a crucial role in determining matrix inverses.
It is essentially the multiplication equivalent of the number 1 for real numbers.
For 2x2 matrices, the identity matrix is represented as \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

• When any square matrix is multiplied by the identity matrix, it remains unchanged.
• It consists of 1's on its diagonal and 0's elsewhere.

In the exercise, to establish that matrices \( R \) and \( S \) are inverses, their product is compared to the identity matrix.
If their product equals the identity matrix, it confirms they are inverses.

2x2 Matrices

2x2 matrices are simple yet powerful arrays used in many mathematical applications.
They consist of two rows and two columns, making them compact and easy to work with.
In terms of linear transformations, they can represent operations like scaling, rotation, and reflections.

• Each element in a 2x2 matrix can be any real number or expression.
• They are often a starting point for learning about matrix operations due to their simplicity.

In this exercise, both matrices \( R \) and \( S \) are 2x2 matrices.
The multiplication of these matrices is performed to determine if they are inverses of each other.

Element Calculation

Element calculation is the heart of matrix multiplication.
It requires a precise approach to ensure accurate results during operations like determining matrix inverses.

• Each element of the resultant matrix is derived by the dot product of a row from the first matrix and a column of the second matrix.
• For a 2x2 matrix, this involves only four elements, minimizing complexity but requiring careful arithmetic.

In the given exercise, element calculation was employed to verify that both \( R \times S \) and \( S \times R \) produce the identity matrix.
This thorough calculation ensures the matrices are exactly inverses.
Each calculated element confirms this by matching the elements of an identity matrix.