Question

In each case either prove the assertion or give an example showing that it is false. a. If \(A \neq 0\) is a square matrix, then \(A\) is invertible. b. If \(A\) and \(B\) are both invertible, then \(A+B\) is invertible. c. If \(A\) and \(B\) are both invertible, then \(\left(A^{-1} B\right)^{T}\) is invertible. d. If \(A^{4}=3 I\), then \(A\) is invertible. e. If \(A^{2}=A\) and \(A \neq 0,\) then \(A\) is invertible. f. If \(A B=B\) for some \(B \neq 0\), then \(A\) is invertible. g. If \(A\) is invertible and skew symmetric \(\left(A^{T}=-A\right)\), the same is true of \(A^{-1}\). h. If \(A^{2}\) is invertible, then \(A\) is invertible. i. If \(A B=I,\) then \(A\) and \(B\) commute.

Short answer

False: Part a, b, e, f, i; True: Part c, d, g, h.

Step-by-step solution

Statement Analysis - Part a

We want to check if a square matrix \( A eq 0 \) implies \( A \) is invertible. A matrix having non-zero determinant is necessary for inversibility. However, there are non-zero square matrices (not identity) with zero determinants, hence they are not invertible. An example of such a matrix is \( \left[ \begin{array}{cc} 1 & 1 \ 1 & 1 \end{array} \right] \), as its determinant is 0.

Statement Analysis - Part b

For matrices \( A \) and \( B \) to be invertible, both must be nonsingular individually. However, \( A+B \) can still be singular (non-invertible). Consider \( A = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \) and \( B = \left[ \begin{array}{cc} -1 & 0 \ 0 & -1 \end{array} \right] \) such that \( A+B = 0 \), which is not invertible.

Statement Analysis - Part c

Given \( A \) and \( B \) are invertible, \( A^{-1} \) and \( B \) are also invertible. The transpose of a product is the product of the transposes in reverse order, so \( (A^{-1} B)^T = B^T (A^{-1})^T \), which is invertible since \( B^T \) and \( (A^{-1})^T \) are invertible.

Statement Analysis - Part d

If \( A^4 = 3I \), then multiplying by \( \frac{1}{3} \) gives \( A^4 \cdot \frac{1}{3} = I \), which implies \( A^3 \) is the inverse of \( 3A \). Thus, \( A \) is invertible.

Statement Analysis - Part e

For \( A^2 = A \), \( A \) can be either invertible or singular. A counterexample is \( A = \left[ \begin{array}{cc} 1 & 0 \ 0 & 0 \end{array} \right] \), which satisfies \( A^2 = A \) but is not invertible.

Statement Analysis - Part f

If \( AB = B \) and \( B eq 0 \), this implies \( A = I \). If \( A \) were not the identity, \( B \) could be zero or such relation holds. But if \( B eq 0 \), \( A \) may still not be invertible.

Statement Analysis - Part g

Given \( A \) invertible and \( A^T = -A \), the inverse \( A^{-1} \) will satisfy \( (A^T)^{-1} = (A^{-1})^T = -A^{-1} \), hence \( A^{-1} \) is also skew-symmetric.

Statement Analysis - Part h

If \( A^2 \) is invertible, then its determinant is non-zero: \( \det(A^2) = (\det A)^2 eq 0 \), implying \( \det A eq 0 \). Hence, \( A \) is invertible.

Statement Analysis - Part i

For \( AB = I \), we know \( A \) and \( B \) are inverse of each other but not necessarily that they commute. A counterexample is \( A = \left[ \begin{array}{cc} 0 & 1 \ 1 & 0 \end{array} \right] \), and \( B = A \), both commute here giving \( AB = BA \), illustrating non-commutativity is possible separately.

Key concepts

Determinants

Understanding determinants is crucial when studying invertibility of matrices. A determinant is a special number assigned to a square matrix that reflects certain properties of the matrix. For a matrix to be invertible, its determinant must be non-zero.

Here’s why: A zero determinant implies that the matrix is singular, meaning it does not have an inverse. For example, consider the matrix \(\left[ \begin{array}{cc} 1 & 1 \ 1 & 1 \end{array} \right] \). Its determinant, calculated as \(1 \cdot 1 - 1 \cdot 1 = 0\), indicates that the matrix is singular and therefore not invertible.

To find the determinant of a 2x2 matrix \(\left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), use the formula \(ad - bc\). Determinants of larger matrices can be found using various methods, such as the Laplace expansion or row reduction.

Inverse Matrices

An inverse matrix essentially reverses the effect of the original matrix. For a matrix \( A \) to have an inverse, it must be square and its determinant must be non-zero. The inverse of a matrix \( A \) is denoted as \( A^{-1} \) and satisfies the equation \( AA^{-1} = A^{-1}A = I \), where \( I \) is the identity matrix.

Calculating the inverse of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) can be done by finding \( \frac{1}{ad-bc} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \), assuming that \( ad-bc eq 0 \). If this determinant is zero, the matrix does not have an inverse, meaning it is singular.

Skew Symmetric Matrices

A skew symmetric matrix is a special type of matrix where the transpose of a matrix is equal to its negative. Mathematically, \( A^T = -A \). Skew symmetric matrices have interesting properties. For instance, all the diagonal elements of a skew symmetric matrix are zero.

If a skew symmetric matrix is invertible, its inverse will also be skew symmetric. This is because if \( A\) is skew symmetric and invertible, \( (A^T)^{-1} = (A^{-1})^T = -A^{-1} \), maintaining the property \((A^{-1})^T = -A^{-1}\). Such matrices are of great interest in various areas of physics and engineering.

Matrix Transposition

Matrix transposition involves flipping a matrix over its diagonal, effectively switching the row and column indices of each element. For example, the transpose of a matrix \( A \) is denoted by \( A^T \). If \( A \) is a matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), then \( A^T \) is \( \left[ \begin{array}{cc} a & c \ b & d \end{array} \right] \).

Transposition is important because the transpose of a product of matrices is equal to the product of their transposes in reverse order, \((AB)^T = B^TA^T\). This property is particularly useful when calculating the invertibility of transposed matrices.

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices by taking the dot product of rows and columns. For matrices \( A \) and \( B \) where the number of columns of \( A \) equals the number of rows of \( B \), their product \( AB \) is defined. The element in the \( i \)-th row, \( j \)-th column of \( AB \) is computed as \( \sum_{k} a_{ik}b_{kj} \).

When both matrices are invertible, their product is also invertible, and \( (AB)^{-1} = B^{-1}A^{-1} \). Matrix multiplication does not generally commute, meaning \( AB eq BA \). This property becomes crucial in problems involving the invertibility and form of matrix products.

Singular Matrices

A singular matrix is a square matrix that does not have an inverse, primarily due to having a determinant of zero. These matrices are significant because they represent systems of equations that do not have a unique solution.

Singular matrices often arise in linear algebra problems where redundancy or linear dependence exists among rows or columns. To determine if a matrix is singular, check if its determinant is zero. If so, the matrix is not invertible, and it is classified as singular. Understanding singular matrices helps in recognizing systems with no or infinitely many solutions in linear algebra.