Problem 72 - College Algebra

Question

Which one of the following is true? \(B,\) and \(A B\) are a. \((A B)^{-1}=A^{-1} B^{-1},\) assuming \(A\) invertible. b. \((A+B)^{-1}=A^{-1}+B^{-1},\) assuming \(A, B,\) and \(A+B\) are invertible. c. \(\left[\begin{array}{rr}1 & -3 \\ -1 & 3\end{array}\right]\) is an invertible matrix. d. None of the above is true.

Step-by-Step Solution

Answer

The correct answer is option 'd'. None of the other options is true.

Step 1: Verify Option 'a'

According to the rules of matrix inversion, it's actually true that \((AB)^{-1} = B^{-1}A^{-1}\), not \((AB)^{-1} = A^{-1}B^{-1}\). So, option 'a' is incorrect.

Step 2: Verify Option 'b'

Option 'b' suggests that the inverse of the sum of two matrices equals the sum of the inverses of the two matrices. This is incorrect. The rules of matrix inversion do not support this claim. So, option 'b' is also incorrect.

Step 3: Verify Option 'c'

For a matrix to be invertible, its determinant should not be equal to zero. The determinant of the given matrix \(\left[\begin{array}{rr}1 & -3 \ -1 & 3\end{array}\right]\) can be calculated as (1*3) - (-1*-3) = 0. Since the determinant is zero, the matrix is not invertible. So, option 'c' is incorrect.

Step 4: Identify the Valid Option

Since all options 'a', 'b', and 'c' were found to be incorrect, the only remaining option is 'd', which is hence the correct answer.

Key Concepts

Matrix Determinant

The determinant of a matrix is a special number that provides key insights into the properties of the matrix. It's particularly important when determining if a matrix is invertible (non-zero determinant) or not (zero determinant). To find the determinant of a 2x2 matrix, use the formula: \[\det\begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc\]This means you multiply the diagonal elements and then subtract the product of the other two elements. For example, in the matrix \(\begin{pmatrix} 1 & -3 \ -1 & 3 \end{pmatrix}\), the determinant is calculated as \(1 \times 3 - (-1) \times (-3) = 0\). Since the determinant is zero, this matrix isn't invertible. This concept is crucial when dealing with questions related to matrix invertibility.

Invertible Matrices

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a matrix that has an inverse. The inverse of a matrix \(A\) is another matrix denoted as \(A^{-1}\), such that the multiplication of \(A\) and \(A^{-1}\) results in the identity matrix \(I\).
To qualify as invertible, a square matrix must have a non-zero determinant. When the determinant is zero, it means the matrix is singular, and thus, it cannot be inverted. This is a vital concept in linear algebra as it determines the possibility of solving linear equations using matrices. In the context of matrix products, remember that \[(AB)^{-1} = B^{-1}A^{-1}\]not \((AB)^{-1} = A^{-1}B^{-1}\). Misunderstanding of these properties often leads to incorrect solutions.

Matrix Operations

Matrix operations include addition, subtraction, multiplication, and finding the inverse, each with its own set of rules. Understanding these operations facilitates dealing with complex problems in linear algebra. Here are some points to remember:

• Addition and Subtraction: Performed element-wise on matrices of the same dimensions.
• Multiplication: Matrix multiplication is not commutative, meaning \(AB eq BA\). Multiply the rows of the first matrix by the columns of the second matrix.
• Inverse: The inverse of a matrix \(A\) exists only if \(A\) is square and has a non-zero determinant.

These operations form the backbone of solving linear equations and transforming geometric data in fields such as physics and computer science. Fully grasping matrix operations helps avoid common pitfalls such as incorrectly applying inverse rules or misapplying matrix multiplication properties.