Question

Let A and B be orthogonal and det A+det B=0. Then

• A. A+B is singular
• B. A+B is non-singular
• C. A+B is orthogonal
• D. A+B is skew-symmetric

Answer 1

The Correct Option is A

Step 1: Analyze the Given Information

We are given that \(AA^T = I\). Taking the determinant of both sides:

\(|AA^T| = |I|\)

\(|A| \cdot |A^T| = 1\)

Since \(|A^T| = |A|\), we have:

\(|A|^2 = 1\)

\(|A| = \pm 1\)

We are also given that \(|A| = 1\).

Step 2: Analyze B

To continue, we must know something that holds true for Matrix B. Let's add information into the equation such as |A|= 1 the |B|=-1 and say that Then the new equation of A+B=0

Step 3: Analyze A + B, if |A|= 1 the |B|=-1 and |A+B|=0 

Add these equation together for the total value, then |A+B|=0 Then A+B is Singular, and can hold true

Conclusion:

Given |A|= 1 the |B|=-1 and |A+B|=0 : A+B is singular.

Answer 2

Step 1: Orthogonal Matrix Property

Since A and B are orthogonal matrices, we have:

\(AA^T = I\) and \(BB^T = I\)

Taking determinants, we get:

\(|AA^T| = |I| = 1\) and \(|BB^T| = |I| = 1\)

\(|A||A^T| = 1\) and \(|B||B^T| = 1\)

Since \(|A^T| = |A|\) and \(|B^T| = |B|\),

\(|A|^2 = 1\) and \(|B|^2 = 1\)

Thus, \(|A| = \pm 1\) and \(|B| = \pm 1\)

Step 2: Using the Given Condition |A| = -|B|

We are given that \(|A| = -|B|\). This means if \(|A| = 1\) then \(|B| = -1\) and if \(|A| = -1\) then \(|B| = 1\). In any case

\(|A||B| = -1\)

Step 3: Analyzing |A+B|

Consider \(|A + B|\). Multiply by the Identity Matrix in a clever way to insert the Orthogonal matrix properties.

\(|A + B| = |A \cdot I + B \cdot I| = |A BB^T + A^T A B|\)

\(=|ABB^T + A^TB||\)

Factoring gives:

\( |A + B| = |A (B^T + A^T)B|\)

Taking Determinants:

\(|A+B| = |A| |B^T+A^T | |B| =|A||B||B^T+A^T |\)

Rearranging and rewritting gives us:

\(=|A||B| (|(A+B)^T|) \)

Since \(|A||B|=-1\) So the solution is given as \(|A+B| = -|(A+B)^T|\) \(|A+B|=-|A+B|\) \(2|A+B|=0\) Then \(|A+B|=0\)

Step 4: Concluding Singularity

Since the determinant of \(A + B\) is 0, \(A + B\) is a singular matrix.