Problem 90 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { If } A=\left[\begin{array}{ll}3 & 5 \\\2 & 4\end{array}\right], \text { find}\left(A^{-1}\right)^{-1}$$

Step-by-Step Solution

Answer

The statement is true. The inverse of the inverse of matrix \(A\) is the matrix \(A\) itself.

Step 1: Understanding the inverse of a matrix

A matrix multiplied by its inverse results in the identity matrix. That is, \(A * A^{-1} = I\). Similarly, \(A^{-1} * A = I\). Here, \(A^{-1}\) is the inverse of matrix \(A\) and \(I\) is the identity matrix.

Step 2: Finding the inverse of the inverse

According to the properties of inverse matrices, the inverse of the inverse of a matrix is the matrix itself. So, in this case, \(\left(A^{-1}\right)^{-1} = A\).

Step 3: Conclusion

Therefore, the statement is true. No modification is needed to make the statement true.

Key Concepts

Identity Matrix Explained

When discussing matrices, the identity matrix is a key component. The identity matrix is the matrix equivalent of number 1 in arithmetic. It's called an "identity" because, when you multiply it with another matrix, it doesn’t change that matrix. This matrix is square-shaped, meaning it has the same number of rows and columns.

• The identity matrix for a 2x2 system looks like this: \[ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]
• This matrix behaves neutrally under multiplication; that is, for any matrix \( A \), multiplying by the identity matrix gives: \( A \times I = A \)

Understanding the identity matrix is crucial because it acts as a starting point when working with inverse matrices. If you know your matrix multiplication well enough, you recognize this special matrix by its zeros and ones along its diagonal.

Inverse Matrix Properties

Inverse matrices can seem complex, but they hold fascinating properties. The inverse of a matrix \( A \) is denoted by \( A^{-1} \). When you multiply \( A \) by its inverse \( A^{-1} \), you end up with the identity matrix \( I \). This property is crucial as it confirms that the two matrices undo each other's operations.
The statement \( A \times A^{-1} = I \) is true for any suitable matrix \( A \). It's also reversible, meaning \( A^{-1} \times A = I \).

• Non-singular requirement: A matrix needs to be non-singular or invertible, meaning it must have a non-zero determinant, to possess an inverse.
• Inverse of the inverse: If you take the inverse of \( A \)'s inverse, like \( (A^{-1})^{-1} \), you're right back where you started with the original matrix \( A \). Thus, \( (A^{-1})^{-1} = A \).

Understanding these properties of inverses helps clarify the relationships between different matrix operations and solidifies your foundation in linear algebra.

Matrix Operations Demystified

Matrix operations form the backbone of solving equations in linear algebra. Working with matrices involves different operations like addition, subtraction, multiplication, and finding inverses.

• Addition/Subtraction: These operations are straightforward. Simply add or subtract corresponding elements. This only works for matrices of the same size.
• Matrix Multiplication: This is a bit more involved. Multiply elements of rows by elements of columns and sum them up. The number of columns in the first matrix must match the number of rows in the second matrix to perform this operation.
• Scalar Multiplication: Multiply each element of the matrix by a scalar (a single number).
• Finding Inverses: Calculating an inverse is a specific operation that involves using the determinant and sometimes adjugates for larger matrices.

By mastering these basic matrix operations, you can effectively handle more complex problems, such as those involving systems of equations and transformations in coordinate geometry.