Analyzing Matrix Properties: Determinant and Inverse of Sums

The question asks us to consider two statements regarding the properties of two non-singular matrices, A and B, of the same order. A non-singular matrix is a square matrix whose determinant is non-zero, and therefore, it has an inverse.

Statement 1: Det (A + B) = det A + det B

This statement claims that the determinant of the sum of two matrices is equal to the sum of their individual determinants. Let's analyze this property using an example.

Consider two 2x2 non-singular matrices:

\(A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

First, let's find the determinants of A and B:

\(\text{det } A = (1 \times 1) - (0 \times 0) = 1\)

\(\text{det } B = (1 \times 1) - (0 \times 0) = 1\)

Now, let's find the sum of the determinants:

\(\text{det } A + \text{det } B = 1 + 1 = 2\)

Next, let's find the sum of the matrices A and B:

\(A + B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1+1 & 0+0 \\ 0+0 & 1+1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

Now, let's find the determinant of the sum (A + B):

\(\text{det } (A + B) = (2 \times 2) - (0 \times 0) = 4 - 0 = 4\)

Comparing det (A + B) with det A + det B:

\(4 \neq 2\)

As shown by this example, det (A + B) is generally not equal to det A + det B. Therefore, statement 1 is incorrect.

Statement 2: (A + B)\(^{-1}\) = A\(^{-1}\) + B\(^{-1}\)

This statement claims that the inverse of the sum of two matrices is equal to the sum of their individual inverses. Let's analyze this property using the same example matrices A and B.

Consider the non-singular matrices:

\(A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

Since A and B are identity matrices, their inverses are themselves:

\(A^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

\(B^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

Now, let's find the sum of the inverses:

\(A^{-1} + B^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1+1 & 0+0 \\ 0+0 & 1+1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

Next, we need to find the inverse of the sum (A + B). We already found A + B:

\(A + B = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

To find the inverse of A + B, we can use the formula for the inverse of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), which is \(\frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\). For A + B = \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\), we have a=2, b=0, c=0, d=2. The determinant is (2)(2) - (0)(0) = 4.

\((A + B)^{-1} = \frac{1}{4} \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 2/4 & 0/4 \\ 0/4 & 2/4 \end{pmatrix} = \begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \end{pmatrix}\)

Comparing (A + B)\(^{-1}\) with A\(^{-1}\) + B\(^{-1}\):

\(\begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \end{pmatrix} \neq \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

As shown by this example, (A + B)\(^{-1}\) is generally not equal to A\(^{-1}\) + B\(^{-1}\). Therefore, statement 2 is incorrect.

Determining the Correctness of Matrix Properties Statements

We have analyzed both statements using examples and found that neither statement holds true in general for non-singular matrices A and B of the same order.

• Statement 1: Det (A + B) = det A + det B is incorrect.
• Statement 2: (A + B)\(^{-1}\) = A\(^{-1}\) + B\(^{-1}\) is incorrect.

Therefore, based on our analysis, neither statement is correct.

Revision Table: Key Matrix Properties

Additional Information: Non-Singular Matrices and Inverses

A square matrix A is called non-singular (or invertible or non-degenerate) if there exists a square matrix B of the same order such that AB = BA = I, where I is the identity matrix. The matrix B is unique and is called the inverse of A, denoted by A\(^{-1}\).

Key facts about non-singular matrices:

• A matrix A is non-singular if and only if det A \(\neq\) 0.
• If A and B are non-singular matrices of the same order, then their product AB is also non-singular, and (AB)\(^{-1}\) = B\(^{-1}\) A\(^{-1}\).
• The sum of two non-singular matrices (A + B) is not necessarily non-singular. For example, if B = -A, and A is non-singular, then B is also non-singular, but A + B = A + (-A) = 0 (the zero matrix), which is singular (det(0) = 0). Even if A+B is non-singular, the properties tested in the question do not hold in general.
• The properties of determinants and inverses are crucial in solving systems of linear equations and in various areas of mathematics, physics, and engineering.