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Magazine Chapter 5: Problem 20
Determine whether the given matrix is orthogonal. If it is, find its inverse. $$\left[\begin{array}{rrrr} \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \end{array}\right]$$
Short Answer
Expert verified
The matrix is orthogonal; its inverse is its transpose.
Step by step solution
01
Definition of Orthogonal Matrix
A matrix is called orthogonal if its transpose is equal to its inverse, meaning that multiplying the matrix by its transpose results in the identity matrix. For a given matrix \( A \), this can be represented as \( A^T A = I \), where \( I \) is the identity matrix.
02
Transpose the Matrix
Calculate the transpose of the given matrix. The transpose of a 4x4 matrix is obtained by interchanging rows and columns. The transpose \( A^T \) of the given matrix \[\begin{bmatrix}\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\end{bmatrix}\] is\[\begin{bmatrix}\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\end{bmatrix}\].
03
Multiply Matrix by its Transpose
Multiply the original matrix by its transpose to check if the result is the identity matrix. Calculation:\[\begin{bmatrix}\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\end{bmatrix}\begin{bmatrix}\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\end{bmatrix}\]Calculating these, you will check all elements in the resulting matrix are:\[\begin{bmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1 \\end{bmatrix}\], which is the identity matrix.
04
Conclude Orthogonality and Inverse
Since \( A A^T = I \), we conclude that the matrix is orthogonal. For an orthogonal matrix, its inverse is equal to its transpose. Therefore, the inverse of the given matrix is:\[\begin{bmatrix}\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\end{bmatrix}\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Transpose
The transpose of a matrix is obtained by swapping its rows with its columns. Imagine flipping the matrix along its diagonal. For a matrix \( A \), the transpose is denoted as \( A^T \). This concept is crucial in specific computations such as determining orthogonality, where the transpose plays a critical role.
The original matrix provided is:
The original matrix provided is:
- \( \left[ \begin{array}{rrrr} \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \end{array} \right] \)
- \( \left[ \begin{array}{rrrr} \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{array} \right] \)
Identity Matrix
An identity matrix is a special kind of square matrix equivalent to one in numeral form. It functionally represents the "do nothing" operation in matrix multiplication, as any matrix multiplied by the identity matrix remains unchanged. The identity matrix is denoted by \( I \) and has \( 1 \)s on its diagonal and \( 0 \) elsewhere.
Consider a 4x4 identity matrix:
Consider a 4x4 identity matrix:
- \( \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1 \end{array} \right] \)
Matrix Multiplication
Matrix multiplication is a way of combining two matrices to produce another matrix. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. Here's how it works:
- Each element \( c_{ij} \) of the product matrix is computed by summing the products of the elements in the i-th row of the first matrix with the corresponding elements in the j-th column of the second matrix.
- \( A \times A^T \)
- The result should be an identity matrix for orthogonal matrices.
Inverse of a Matrix
The inverse of a matrix \( A \), denoted as \( A^{-1} \), is a matrix that when multiplied by \( A \) yields the identity matrix: \( AA^{-1} = I \) and \( A^{-1}A = I \).
Finding the inverse can generally be a complex task, but it greatly simplifies if the matrix is orthogonal.
Finding the inverse can generally be a complex task, but it greatly simplifies if the matrix is orthogonal.
- For orthogonal matrices, the inverse is easier to determine.
- Interestingly, the inverse of an orthogonal matrix is simply its transpose: \( A^{-1} = A^T \).
