Understanding Orthogonal Matrices and Matrix Properties

This question asks us to evaluate two statements about matrix multiplication and determinants involving an orthogonal matrix A and a specific matrix B.

First, let's recall what an orthogonal matrix is and its key properties.

What is an Orthogonal Matrix?

An orthogonal matrix is a square matrix whose transpose is equal to its inverse. If A is an orthogonal matrix, then:

• \(A^T = A^{-1}\)
• Consequently, \(A^T A = I\) and \(A A^T = I\), where I is the identity matrix.

A crucial property related to the determinant of an orthogonal matrix can be derived from \(A^T A = I\). Taking the determinant of both sides:

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

We know that the determinant of a product of matrices is the product of their determinants: \(|AB| = |A| |B|\). Also, the determinant of a transpose is equal to the determinant of the original matrix: \(|A^T| = |A|\). The determinant of the identity matrix is 1.

So, \(|A^T| |A| = 1\), which means \(|A| |A| = 1\), or \(|A|^2 = 1\).

Therefore, the determinant of an orthogonal matrix is always either 1 or -1: \(|A| = \pm 1\).

Evaluating Statement 1: Determinant of AB

Statement 1 claims that \(|AB| = \pm 47\). We can use the property \(|AB| = |A| |B|\).

We know \(|A| = \pm 1\) for an orthogonal matrix A.

Now, let's calculate the determinant of the given matrix B:

\({\rm{B}} = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ { - 3}&0&2\\ 2&5&0 \end{array}} \right]\)

We can compute the determinant by expanding along the first row:

\(|B| = 1 \times \det \left[ {\begin{array}{*{20}{c}} 0&2\\ 5&0 \end{array}} \right] - 2 \times \det \left[ {\begin{array}{*{20}{c}} { - 3}&2\\ 2&0 \end{array}} \right] + 3 \times \det \left[ {\begin{array}{*{20}{c}} { - 3}&0\\ 2&5 \end{array}} \right]\)

\(|B| = 1 \times (0 \times 0 - 2 \times 5) - 2 \times ((-3) \times 0 - 2 \times 2) + 3 \times ((-3) \times 5 - 0 \times 2)\)

\(|B| = 1 \times (0 - 10) - 2 \times (0 - 4) + 3 \times (-15 - 0)\)

\(|B| = 1 \times (-10) - 2 \times (-4) + 3 \times (-15)\)

\(|B| = -10 + 8 - 45\)

\(|B| = -2 - 45\)

\(|B| = -47\)

Now, we can find \(|AB|\):

\(|AB| = |A| |B|\)

\(|AB| = (\pm 1) \times (-47)\)

\(|AB| = \mp 47\), which is equivalent to \(\pm 47\).

So, Statement 1, \(|AB| = \pm 47\), is correct.

Evaluating Statement 2: Matrix Commutation AB = BA

Statement 2 claims that AB = BA.

Matrix multiplication is generally not commutative. That is, for two matrices A and B, AB is typically not equal to BA. For AB = BA to hold, the matrices A and B must commute. There is no general property stating that an orthogonal matrix A will commute with an arbitrary matrix B.

The statement AB = BA must hold for *any* orthogonal matrix A of order 3 and the given matrix B for Statement 2 to be correct. Since matrix multiplication is not commutative in general, and there is no special relationship given between all 3x3 orthogonal matrices and the specific matrix B that would force them to commute, this statement is incorrect.

For example, let A be a simple orthogonal matrix like the identity matrix \(I_3\). \(I_3\) is orthogonal because \(I_3^T = I_3\) and \(I_3 I_3 = I_3\). In this case, \(AB = I_3 B = B\) and \(BA = B I_3 = B\), so AB = BA holds. However, the statement must hold for *any* orthogonal matrix. Consider a rotation matrix in 3D (which is an orthogonal matrix). Multiplying a general matrix B by a rotation matrix A will generally result in a different matrix than multiplying A by B in the reverse order.

Therefore, Statement 2, AB = BA, is incorrect.

Conclusion based on Statement Evaluation

Based on our analysis:

• Statement 1: \(|AB| = \pm 47\) is correct.
• Statement 2: AB = BA is incorrect.

Thus, only Statement 1 is correct.

Revision Table: Key Matrix Properties

Additional Information: Applications of Orthogonal Matrices

Orthogonal matrices have many important applications in various fields, especially where transformations that preserve length and angles are involved. Some key areas include:

• Geometry and Transformations: Rotation, reflection, and scaling transformations in linear algebra can be represented by orthogonal matrices. These transformations preserve the magnitude of vectors and the angle between them.
• Numerical Analysis: Orthogonal matrices are used in algorithms for matrix decomposition (like QR decomposition) which are fundamental in solving linear systems, eigenvalue problems, and least squares problems.
• Quantum Mechanics: Unitary matrices (the complex equivalent of orthogonal matrices) are used to represent the evolution of quantum states, as they preserve probability amplitudes.
• Signal Processing and Data Analysis: Techniques like Principal Component Analysis (PCA) involve transformations represented by orthogonal matrices to decorrelate data while preserving variance.
• Computer Graphics: Orthogonal matrices are widely used for transformations in 3D graphics pipelines.

The property \(|A| = \pm 1\) for an orthogonal matrix A implies that such transformations preserve volume. If \(|A|=1\), the transformation corresponds to a rotation (preserving orientation). If \(|A|=-1\), it corresponds to a reflection possibly combined with rotation (reversing orientation).