Problem 6 - College Algebra

Question

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$A=\left[\begin{array}{rr}1 & 0 \\ -1 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

Matrix A is the inverse of matrix B.

Step 1: Understand the Definition of Inverse Matrices

Two matrices are inverses of each other if their product is the identity matrix. For 2x2 matrices, the identity matrix is \( I = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \). We need to show \( AB = I \) and \( BA = I \).

Step 2: Calculate the Product AB

To find \( AB \), multiply matrix \( A \) by matrix \( B \): \[AB = \left[ \begin{array}{rr} 1 & 0 \ -1 & 1 \end{array} \right] \left[ \begin{array}{ll} 1 & 0 \ 1 & 1 \end{array} \right] = \left[ \begin{array}{rr} (1)(1)+(0)(1) & (1)(0)+(0)(1) \ (-1)(1)+(1)(1) & (-1)(0)+(1)(1) \end{array} \right] = \left[ \begin{array}{rr} 1 & 0 \ 0 & 1 \end{array} \right].\]

Step 3: Show AB Equals I

After calculating \( AB \), we found that it equals the identity matrix, \( I = \left[ \begin{array}{rr} 1 & 0 \ 0 & 1 \end{array} \right] \). Therefore, \( AB = I \).

Step 4: Calculate the Product BA

Now, calculate \( BA \) by multiplying \( B \) by \( A \): \[BA = \left[ \begin{array}{ll} 1 & 0 \ 1 & 1 \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \ -1 & 1 \end{array} \right] = \left[ \begin{array}{rr} (1)(1)+(0)(-1) & (1)(0)+(0)(1) \ (1)(1)+(1)(-1) & (1)(0)+(1)(1) \end{array} \right] = \left[ \begin{array}{rr} 1 & 0 \ 0 & 1 \end{array} \right].\]

Step 5: Show BA Equals I

After computing \( BA \), we confirm that it also equals the identity matrix, \( I = \left[ \begin{array}{rr} 1 & 0 \ 0 & 1 \end{array} \right] \). Thus, \( BA = I \).

Step 6: Conclusion

Since both \( AB \) and \( BA \) equal the identity matrix \( I \), matrix \( A \) is indeed the inverse of matrix \( B \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra, where the elements of two matrices are combined to produce a new matrix. When multiplying two matrices, it is important to ensure that the number of columns in the first matrix matches the number of rows in the second matrix.
For example, consider matrices \( A \) and \( B \). To find the product \( AB \), you follow these steps:

• Multiply each element of the rows of \( A \) by the corresponding element of the columns of \( B \).
• Sum the products for each corresponding row and column pair.
• This sum becomes the element of the resulting matrix \( AB \) at the position that corresponds to the row number of \( A \) and the column number of \( B \).

Matrix multiplication is not commutative, meaning \( AB \) might not equal \( BA \). However, it is associative and distributive over addition.

Identity Matrix

The identity matrix is a special kind of matrix that acts as a multiplicative identity in matrix algebra, fulfilling a similar role to the number '1' in basic arithmetic. For any matrix \( A \) of size \( n \times n \), when it is multiplied by the identity matrix \( I \), the result is \( A \) itself:

• For example, if \( A \) is a 2x2 matrix, \( I \) is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).
• The identity matrix does not change \( A \) when multiplying: \( AI = IA = A \).

The concept of an identity matrix is crucial when discussing inverse matrices, as two matrices are inverses of each other if their product is the identity matrix.

2x2 Matrices

A 2x2 matrix is one of the simplest forms of matrices, having 2 rows and 2 columns. This makes them convenient for introducing concepts in matrix algebra:

• A general 2x2 matrix \( M \) can be represented as \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \).
• The determinant of a 2x2 matrix \( M \), calculated as \( ad - bc \), plays an important role in determining if the matrix has an inverse.
• If the determinant is not zero, the matrix is invertible, meaning it has an inverse matrix.

Understanding basic operations and properties of 2x2 matrices helps in grasping more complex ideas in matrix algebra.

Matrix Algebra

Matrix algebra involves various operations and theories related to matrices, making it a powerful tool in mathematics and applied sciences. Here's a rundown of essential concepts:

• Matrix addition and subtraction align corresponding elements of matrices of the same size.
• Scalar multiplication involves multiplying each element of a matrix by a scalar (a real number).
• Matrix division is not directly possible, but we solve equations involving matrices using inverse matrices.
• The inverse of a matrix is used to find solutions to matrix equations, where we have \( AX = B \). By multiplying both sides by \( A^{-1} \), if it exists, we isolate \( X \) (i.e., \( X = A^{-1}B \)).

These foundational operations enable more complex transformations and solutions involving linear equations and vector spaces. Through matrix algebra, you gain insights into solving real-world problems.