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Chapter 5: Problem 51

If \(A\) is a non-singular matrix, then (A) \(A^{-1}\) is symmetric if \(A\) is symmetric (B) \(A^{-1}\) is skew-symmetric if \(A\) is symmetric (C) \(\left|A^{-1}\right|=|A|\) (D) \(\left|A^{-1}\right|=|A|^{-1}\)

Short Answer

Expert verified
Options (A) and (D) are correct; (B) and (C) are incorrect.

Step by step solution

01

Understanding Matrix Inverse Property

The inverse of a matrix \(A\), denoted \(A^{-1}\), has a property where \(A \times A^{-1} = I\), where \(I\) is the identity matrix of the same dimension as \(A\). This is fundamental in solving problems involving inverse matrices.
02

Evaluate Symmetric Matrix Inversion

A matrix \(A\) is symmetric if \(A^T = A\). If \(A\) is symmetric and non-singular, then \(A^{-1}\) being symmetric implies \((A^{-1})^T = A^{-1}\). This is true because \[ (A^{-1})^T = (A^T)^{-1} = (A)^{-1}, \] thereby verifying option (A): \(A^{-1}\) is symmetric if \(A\) is symmetric.
03

Evaluate Skew-Symmetric Case

A skew-symmetric matrix \(A\) satisfies \(A^T = -A\). If \(A\) is symmetric, it cannot be skew-symmetric without being a zero matrix. Hence, \(A^{-1}\) being skew-symmetric when \(A\) is symmetric is invalid. Thus, option (B) is incorrect.
04

Determinant of an Inverse Matrix

The determinant of the inverse of matrix \(A\) is given by \[\left|A^{-1}\right| = \frac{1}{|A|}. \]Thus, option (D): \(\left|A^{-1}\right| = |A|^{-1}\) is correct because for non-singular matrices, \( |A| eq 0 \), making the inverse determinant well-defined.
05

Evaluate Option (C)

The statement in option (C) that \[ \left|A^{-1}\right| = |A| \]is incorrect, as we鈥檝e established that \[ \left|A^{-1}\right| = |A|^{-1},\] except when \(|A|=1\). Thus, option (C) is not generally true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetric Matrix
A symmetric matrix is one of the most fundamental concepts in linear algebra. Simply put, a matrix is symmetric if it is equal to its transpose. Let's denote a matrix as \( A \). If \( A^T = A \), then \( A \) is symmetric. This property simplifies many matrix operations and is prevalent in real-world applications, including physics and engineering.
If you're working with symmetric matrices, it's helpful to remember:
  • Symmetric matrices are always square.
  • Eigenvalues of symmetric matrices are always real numbers.
  • If a symmetric matrix is also non-singular, its inverse is also symmetric.
Understanding these characteristics can immensely help in comprehending the behaviors and properties related to matrices, such as when discussing matrix inversion or spectral theorems.
Skew-Symmetric Matrix
In contrast to symmetric matrices, a matrix is skew-symmetric if the transpose of the matrix is the negative of the matrix itself. For a matrix \( A \), let's say \( A^T = -A \). This means that the diagonal elements of a skew-symmetric matrix must necessarily be zero, a unique trait of skew-symmetric matrices.
When engaging with skew-symmetric matrices, these points are essential:
  • All the diagonal elements are zero.
  • Odd-order skew-symmetric matrices are singular, meaning they don't have an inverse.
  • They are useful in various applications, including differentiation operations and physics.
Knowing these distinctive properties can give you valuable insights, especially when dealing with systems where symmetry and anti-symmetry properties are critical.
Determinants
Determinants play a vital role when we determine whether a matrix is invertible, among other applications. The determinant is a scalar value that is a product of a matrix's eigenvalues. It serves as a gatekeeper of sorts because if \( |A| = 0 \), then \( A \) is singular and does not have an inverse.
Key facts about determinants to keep in mind:
  • The determinant allows for the linear independence of a matrix's columns and rows.
  • For a 2x2 matrix, the determinant is calculated as \( ad - bc \).
  • Understanding determinants are crucial for matrix inversion, as the inverse of a matrix \( A \) is \( \frac{1}{|A|} \, times \, the \, adjugate \, of \, A \), provided \( |A| ot= 0 \).
Determinants not only describe the invertibility but also say a lot about the geometric properties of the matrix.
Non-Singular Matrix
A non-singular matrix is one that is invertible, meaning it has an inverse matrix. For a matrix \( A \), there exists the inverse matrix \( A^{-1} \) such that \( A \times A^{-1} = I \), where \( I \) is the identity matrix. Non-singular matrices are crucial in various mathematical computations, including solving systems of linear equations.
Key aspects to know about non-singular matrices:
  • The determinant \( |A| eq 0 \) is a fundamental condition for a matrix to be non-singular.
  • If a matrix is non-singular, its linear system of equations has a unique solution.
  • Non-singular matrices are often contrasted with singular matrices, which do not possess an inverse.
Understanding non-singular matrices can greatly enhance your ability to perform complex matrix operations and appreciate the underlying theory of linear algebra.

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Most popular questions from this chapter

If the product of the matrix \(B=\left[\begin{array}{ccc}2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & 1 & -1\end{array}\right]\) with a matrix \(A\) has inverse \(C=\left[\begin{array}{ccc}-1 & 0 & 1 \\ 1 & 1 & 3 \\\ 2 & 0 & 2\end{array}\right]\), then \(A^{-1}\) equals (A) \(\left[\begin{array}{ccc}-3 & -5 & 5 \\ 0 & 9 & 14 \\ 2 & 2 & 6\end{array}\right]\) (B) \(\left[\begin{array}{ccc}-3 & 5 & 5 \\ 0 & 0 & 9 \\ 2 & 14 & 16\end{array}\right]\) (C) \(\left[\begin{array}{ccc}-3 & -5 & -5 \\ 0 & 9 & 2 \\ 2 & 14 & 6\end{array}\right]\) (D) \(\left[\begin{array}{ccc}-3 & -3 & -5 \\ 0 & 9 & 2 \\ 2 & 14 & 6\end{array}\right]\)

If \(A\) is a square matrix, \(B\) is a singular matrix of same order, then for a positive integer \(n,\left(A^{-1} B A\right)^{n}\) equals (A) \(A^{-\pi} B^{n} A^{n}\) (B) \(A^{n} B^{n} A^{-n}\) (C) \(A^{-1} B^{n} A\) (D) \(n\left(A^{-1} B A\right)\)

Let \(A=\left(\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right) .\) The only correct statement about the matrix \(A\) is (A) \(A\) is a zero matrix (B) \(A^{2}=I\) (C) \(A^{-1}\) does not exist (D) \(A=(-1) I\), where \(I\) is a unit matrix

If \(A=\left[\begin{array}{cc}\alpha & 2 \\ 2 & \alpha\end{array}\right]\) and \(\left|A^{3}\right|=125\) then the value of \(\alpha\) is (A) \(\pm 1\) (B) \(\pm 2\) (C) \(\pm 3\) (D) \(\pm 5\)

If \(A\) and \(B\) are two matrices such that \(A B=B A\), then \(\forall n \in N\) (A) \(A^{n} B=B A^{n}\) (B) \((A B)^{n}=A^{n} B^{n}\) (C) \((A+B)^{n}={ }^{n} C_{0} A^{n}+{ }^{n} C_{1} A^{n-1} B+{ }^{n} C_{2} A^{n-2} B^{2}+\ldots+\) \({ }^{n} C_{n} B^{n}\) (D) \(A^{2 n}-B^{2 n}=\left(A^{n}-B^{n}\right)\left(A^{n}+B^{n}\right)\)
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