Question

Construct an orthogonal matrix using the skew-symmetric matrix (a) \(\left|\begin{array}{cl}-(3 / 5) & -(4 / 5) \\ (4 / 5) & -(3 / 5)\end{array}\right|\) (c) \(\left|\begin{array}{ll}(4 / 5) & (3 / 5) \\ (3 / 5) & (4 / 5)\end{array}\right|\) $$ \text { (d) }\left|\begin{array}{ll} -(4 / 5) & -(3 / 5) \\ -(3 / 5) & -(4 / 5) \end{array}\right| $$

Short answer

The orthogonal matrices constructed using the given skew-symmetric matrices are: (a) \(B =\) \(\left|\begin{array}{cc} \frac{3}{5} & \frac{4}{5} \\ -\frac{4}{5} & \frac{3}{5} \end{array}\right|\) (c) \(B =\) \(\left|\begin{array}{cc} \frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & -\frac{4}{5} \end{array}\right|\) (d) \(B =\) \(\left|\begin{array}{cc} \frac{4}{5} & \frac{-3}{5} \\ \frac{3}{5} & \frac{4}{5} \end{array}\right|\)

Step-by-step solution

(Step 1): Checking whether the given matrices are skew-symmetric

(For each of the given matrices, transpose it and check if it is equal to the negative of the original matrix to confirm they are skew-symmetric.)

(Step 2): Constructing the orthogonal matrix using skew-symmetric matrices

(Since the matrix A satisfies the skew-symmetric condition, we can construct the orthogonal matrix B by applying the condition \(B^T B = I\). We have \(B^T = -A\), so \((-A) B = I\). We can get the orthogonal matrix B by multiplying A and the inverse of \(-A\).)

(Step 3): Calculate the inverse of \(-A\)

(To calculate the inverse of a 2x2 matrix, we swap the elements in the main diagonal, change the signs of the other elements, and divide by the determinant. Apply this method to \(-A\) for each skew-symmetric matrix.)

(Step 4): Multiply A with the inverse of \(-A\)

(Multiply each skew-symmetric matrix A by the inverse of \(-A\) calculated in step 3 to obtain the orthogonal matrix B.) Applying these steps, we have the following orthogonal matrices: (a) \(B =\) \(\left|\begin{array}{cc} \frac{3}{5} & \frac{4}{5} \\ -\frac{4}{5} & \frac{3}{5} \end{array}\right|\) (c) \(B =\) \(\left|\begin{array}{cc} \frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & -\frac{4}{5} \end{array}\right|\) (d) \(B =\) \(\left|\begin{array}{cc} \frac{4}{5} & \frac{-3}{5} \\ \frac{3}{5} & \frac{4}{5} \end{array}\right|\)

Key concepts

Skew-Symmetric Matrices

A skew-symmetric matrix is a special kind of square matrix where the transpose of the matrix is equal to the negative of the original matrix. In simpler terms, if you flip a matrix over its diagonal and multiply it by -1, you should get the initial matrix back. This means:

• Every diagonal element in a skew-symmetric matrix is zero.
• The element at position (i, j) is the negative of the element at position (j, i).

For example, in the matrix \(\begin{array}{cc}-(3 / 5) & -(4 / 5) \ (4 / 5) & -(3 / 5)\end{array}\), you'll notice:
The diagonal elements are zero because a skew-symmetric matrix's diagonal must always be zero.
The element in the first row and second column is the negative of the element in the second row and first column, which confirms it's skew-symmetric.

Matrix Inverse

Finding the inverse of a matrix is like finding the number that a given number can be multiplied by to get one. For matrices, this means finding another matrix that you can multiply your original matrix by to get the identity matrix:

• The identity matrix is like the number one for matrices.
• It has ones on the diagonal and zeros everywhere else.

To find the inverse of a 2x2 matrix, you need to:
Swap the elements of the main diagonal
Change the signs of the off-diagonal elements
Divide each element of the matrix by the matrix's determinant.
For instance, if we consider matrix \(-A\), following these steps correctly will help us find \(A^{-1}\), essential for constructing orthogonal matrices.

Matrix Transpose

Transposing a matrix means flipping it over its diagonal, which swaps the rows and columns.

• The transpose of a matrix is denoted by \(A^T\).
• It's a common operation in linear algebra and is often used when dealing with skew-symmetric matrices.

For instance, if we have:
\(\begin{array}{cc} a & b \ c & d \end{array}\)
The transpose would be:
\(\begin{array}{cc} a & c \ b & d \end{array}\)
In the context of skew-symmetric matrices, the transpose helps verify if the matrix equals its negative counterpart, a defining feature of skew-symmetric matrices.

Determinant Calculation

The determinant of a matrix is like a unique number associated with a matrix that provides a lot of information about the matrix, such as whether a matrix is invertible or not.

• The determinant of a 2x2 matrix \(\begin{array}{cc} a & b \ c & d \end{array}\) is computed as \(ad - bc\).
• If the determinant is non-zero, the matrix is invertible, meaning you can find a matrix that will multiply with it to yield the identity matrix.

In our exercise, calculating the determinant is crucial, as it is used in the process to find the inverse of a matrix.
Once the determinant is known, it aids in applying the inverse formula and further facilitates the construction of orthogonal matrices by combining the skew-symmetric matrix with its inverse.