Question

Which of the following statement (s) is true? (1) Inverse of a square matrix is not unique. (2) If \(\mathrm{A}\) and \(\mathrm{B}\) are two square matrices, then \((\mathrm{AB})^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}} \mathrm{B}^{\mathrm{T}}\) (3) If \(\mathrm{A}\) and \(\mathrm{B}\) are two square matrices, then \((\mathrm{AB})^{-1}=\mathrm{A}^{-1} \mathrm{~B}^{-1}\). (4) If \(A\) is a non-singular square matrix, then its inverse can be uniquely expressed as sum of a symmetric and a skew-symmetric matrix.

Short answer

a. The inverse of a square matrix is not unique. b. If A and B are two square matrices, then \((AB)^T = A^TB^T\). c. If A and B are two square matrices, then \((AB)^{-1} = A^{-1}B^{-1}\). d. If A is a non-singular square matrix, then its inverse can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix. Answer: b. If A and B are two square matrices, then \((AB)^T = A^TB^T\).

Step-by-step solution

Statement 1

The given statement states that the inverse of a square matrix is not unique. According to the definition and property of invertible matrices, if a square matrix A has an inverse, then it is unique. Therefore, statement 1 is false.

Statement 2

The statement says that if A and B are two square matrices, then \((AB)^T = A^TB^T\). Let us check this by transposing the product AB: \((AB)^T = (A_1B_1 + A_2B_2 + \cdots + A_nB_n)^T = B_1^TA_1^T + B_2^TA_2^T + \cdots + B_n^TA_n^T = B^TA^T\) Hence, statement 2 is true.

Statement 3

The statement states that if A and B are two square matrices, then \((AB)^{-1} = A^{-1}B^{-1}\). According to the property of inverse of a product: \((AB)^{-1} = B^{-1}A^{-1}\) This is not the same as the given statement, as the order of the inverses is different. Therefore, statement 3 is false.

Statement 4

The statement says that if A is a non-singular square matrix, then its inverse can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix. Let A be a non-singular square matrix and let S and K be symmetric and skew-symmetric matrices, respectively such that, \(A^{-1} = S + K\) Now, let's calculate the transpose of \(A^{-1}\): \((A^{-1})^T = (S + K)^T\) Since the sum of the transposes of two matrices is equal to the transpose of their sum, and S is symmetric and K is skew-symmetric, we have: \((A^{-1})^T = S^T + K^T = S - K\) Now, if we can prove that \((A^{-1})^T\) is equal to \(A^{-1}\); there is only one possible solution for S and K. So, \(A^{-1} = (A^{-1})^T = S - K\) \(A^{-1} = S + K\) Subtracting the two equations, we get: \(0 = 2K\) Hence, \(K = 0\), which contradicts our assumption that K is a skew-symmetric matrix. Therefore, statement 4 is false. In conclusion, the only true statement among the given options is statement 2.

Key concepts

Inverse of a Square Matrix

Understanding the concept of the inverse of a square matrix is foundational in linear algebra. An inverse of a matrix A, denoted as A^-1, is a matrix that, when multiplied with the original matrix, yields the identity matrix. This process can be likened to the multiplicative inverse of a number, where multiplying the number by its inverse gives 1. The key property here is that the inverse is unique if it exists; in other words, a square matrix cannot have more than one inverse.

Calculating the inverse involves several methods such as the adjugate matrix and elementary row operations. It is essential to remember that not all square matrices have inverses. Those that do are called invertible or non-singular matrices. The process of finding the inverse is deeply tied to the matrix's determinant. If the determinant of a square matrix is zero, it is said to be singular, and it does not possess an inverse. The existence of the inverse is pivotal for solving systems of linear equations and for matrix division, which is not directly possible.

Matrix Transposition

When we transpose a matrix, denoted by the superscript 'T', we flip it over its diagonal, effectively converting row elements into column elements, and vice-versa. For instance, the first row of the original matrix becomes the first column of its transpose. This applies to any M x N matrix, not just square matrices.

Matrix transposition plays a vital role in numerous mathematical operations and properties. For square matrices, when we multiply two matrices together, the transposition of the product is equal to the product of the transposes in the reverse order, that is, \( (AB)^{T} = B^{T}A^{T} \). This property is very useful, especially when dealing with orthogonal or unitary matrices where the transpose of a matrix is also its inverse.

Properties of Matrix Products

The properties of matrix products are akin to the rules we observe in arithmetic but with distinct differences due to the nature of matrices. For instance, matrix multiplication is not commutative, meaning that AB is not necessarily equal to BA.

Another key property is the associative nature of matrix multiplication. Expanding on this, for any three matrices A, B, and C, the product A(BC) is the same as the product (AB)C. Furthermore, when considering the inverse of a matrix product as seen in the exercise, it's essential to note that the order of the inverses is switched: \( (AB)^{-1} = B^{-1}A^{-1} \). This inverse property is powerful when solving linear systems or inverting transformation matrices in various applications.

Symmetric and Skew-Symmetric Matrices

A matrix is symmetric if it is equal to its own transpose, that is, A = A^T. Conversely, a matrix is skew-symmetric if its transpose is equal to the negative of itself, written as A = -A^T. For any square matrix, you can actually decompose it into the sum of a symmetric and a skew-symmetric matrix. The application of these types of matrices is widespread, from mechanics to geometry.

In our exercise, however, there was a misconception. While you can express any square matrix as a sum of a symmetric and a skew-symmetric matrix, the inverse of a non-singular matrix doesn't follow the same rule due to the unique nature of inverses. The fact that an inverse is unique prevents it from being expressed as a sum in such a manner, which was incorrectly implied in one of the exercise statements.

Non-Singular Square Matrix

A non-singular square matrix holds a special place because it signifies that the matrix can be inverted. In simpler terms, for a square matrix A, if the determinant of A is non-zero, it possesses an inverse. This characteristic of being non-singular means that the system of equations corresponding to the matrix has a unique solution.

Such matrices are of particular interest since they validate the existence of linear transformations that are reversible. In practical applications, non-singular matrices are indispensable for algorithms requiring matrix inversion, like finding the solution to a set of linear equations or performing coordinate transformations. Therefore, the distinction between non-singular and singular matrices is pivotal in determining the solvability and behavior of a linear system.