Orthogonal matrix

Square matrix

The square matrix is the matrix in which the number of rows = number of columns. So a matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be a square matrix when $\mathrm{m}=\mathrm{n}$.
E.g.

$
\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]_{3 \times 3} \text { or, } \quad\left[\begin{array}{cc}
2 & -4 \\
7 & 3
\end{array}\right]_{2 \times 2}
$

Orthogonal matrix

A matrix is said to be an orthogonal matrix if the product of a matrix and its transpose gives an identity matrix. A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

A square matrix is said to be an orthogonal matrix if AA’ = I, where I is the identity matrix.

Properties of Orthogonal matrix

1) $A A^{\prime}=I \Rightarrow A^{-1}=A$
2) The product of two orthogonal matrices is also an orthogonal matrix. If $A$ and $B$ are orthogonal then $A B$ is also orthogonal.
3) The inverse of the orthogonal matrix is also orthogonal. If $A$ is orthogonal the $A^{-1}$ is also orthogonal.
4) All the orthogonal matrices are invertible.
5) The determinant of an orthogonal matrix is always equal to the -1 or + 1 . If $A$ is orthogonal then $|A|=1$ or -1
6) All orthogonal matrices are square matrices but not all square matrices are orthogonal.
7) All identity matrices are orthogonal matrices.
8) The transpose of the orthogonal matrix is also orthogonal. If $A$ is orthogonal then $A^{\prime}$ is also orthogonal.

Summary

Orthogonal matrices are a special type of matrices. Orthogonal matrices have the ability to preserve lengths and angles during transformations like rotations and reflections. They are fundamental in fields such as geometry, signal processing, and quantum mechanics, where their properties play a key role in both theoretical understanding and practical applications.

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Solved Examples Based on Orthogonal Matrices

Example 1: A is a orthogonal matrix where $A=\left[\begin{array}{cc}5 & 5 \alpha \\ 0 & \alpha\end{array}\right]$ . Then find the value of .

1) 1
2) $\frac{1}{5}$
3) $\frac{1}{25}$
4) None of these

Solution:
Orthogonal matrix -

$
A A^{\prime}=I
$

- wherein
$A^{\prime}$ is transpose matrix of matrix $A$ and $I$ is identity matrix
Orthogonal matrix

$
A A^T=I, A^T=\left[\begin{array}{cc}
5 & 0 \\
5 \alpha & \alpha
\end{array}\right] A A^T=\left[\begin{array}{cc}
5 & 5 \alpha \\
0 & \alpha
\end{array}\right]\left[\begin{array}{cc}
5 & 0 \\
5 \alpha & \alpha
\end{array}\right]
$
$

\begin{aligned}
& =\left[\begin{array}{cc}
25\left(1+\alpha^2\right) & 5 \alpha^2 \\
5 \alpha^2 & \alpha^2
\end{array}\right] \\
& =\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]
\end{aligned}
$

No value of 2 exist
Example 2: If $\mathrm{A}=\left[\begin{array}{ccc}\frac{1}{3} & \frac{2}{3} & a \\ \frac{2}{3} & \frac{1}{3} & b \\ \frac{2}{3} & -\frac{2}{3} & c\end{array}\right]_{\text {is orthogonal, then find } \mathrm{a}, \mathrm{b}, \mathrm{c}}$
1) $\left( \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{2}{3}\right)$
2) $\left( \pm \frac{2}{3}, \pm \frac{1}{3}, \pm \frac{1}{3}\right)$
3) $\left( \pm \frac{2}{3}, \pm \frac{2}{3}, \pm \frac{1}{3}\right)$
4) $\left( \pm \frac{2}{3}, \pm \frac{1}{3}, \pm \frac{2}{3}\right)$

Example 3: Which of the following statements about an orthogonal matrix \( A \) is **not** true?

1) \( A^{-1} = A^T \)
2) The columns of \( A \) are orthonormal vectors.
3) The determinant of \( A \) is always zero.
4) \( A A^T = I \)

Solution:
1) True. For an orthogonal matrix \( A \), \( A^{-1} = A^T \).
2) True. The columns (and rows) of an orthogonal matrix are orthonormal vectors.
3) False. The determinant of an orthogonal matrix is \( \pm 1 \), not zero.
4) True. For an orthogonal matrix, \( A A^T = I \).

Hence, the answer is option 3.

Solution: We know that Orthogonal matrix - $A A^{\prime}=I$ where, $A^{\prime}$ is a transpose matrix of matrix $A$ and $I$ is the identity matrix

For Orthogonal Matrices, $A A^{\prime}=1$

$
\left[\begin{array}{ccc}
\frac{1}{3} & \frac{2}{3} & a \\
\frac{2}{3} & \frac{1}{3} & b \\
\frac{2}{3} & -\frac{2}{3} & c
\end{array}\right]\left[\begin{array}{ccc}
\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\
\frac{2}{3} & \frac{1}{3} & -\frac{2}{3} \\
a & b & c
\end{array}\right]=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
$

On comparing

$
\begin{aligned}
& -\frac{1}{-9}+\stackrel{4}{-9}+a^2=1 ;-\frac{2}{-9}+\frac{2}{-9}+a b=0 \\
& \Rightarrow a^2=\frac{4}{9} \Rightarrow a= \pm \frac{2}{3} \text { and } b=\mp-3 \text { and } \| l y c= \pm \frac{1}{3}
\end{aligned}
$

Hence, the answer is option 3.
Example 3: Which of the following statements about an orthogonal matrix $\backslash(A \backslash)$ is **not** true?
1) $\backslash\left(A^{\wedge}\{-1\}=A^{\wedge} T \backslash\right)$
2) The columns of $\backslash(A \backslash)$ are orthonormal vectors.
3) The determinant of $\backslash(A \backslash)$ is always zero.
4) $\backslash\left(A A^{\wedge} T=I \backslash\right)$

Solution:
1) True. For an orthogonal matrix $\backslash(A \backslash), \backslash\left(A^{\wedge}\{-1\}=A^{\wedge} T \backslash\right)$.
2) True. The columns (and rows) of an orthogonal matrix are orthonormal vectors.
3) False. The determinant of an orthogonal matrix is $\(\backslash p m 1 \backslash)$ not zero.
Hence the correct option is 3.

Example 4: Suppose that $a, b$, and $c$ are real numbers such that $a+b+c=1$. If the $A=\left|\begin{array}{ccc}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is orthogonal, then:
1) At least one of $a, b$, and $c$ is negative
2) $|\mathrm{A}|$ is negative
3) $a^3+b^3+c^3-3 a b c=1$
4) All of these

$
\begin{aligned}
& \quad\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|=-\left(a^3+b^3+c^3-3 a b c\right) \\
& \text { eg:- } \mathrm{AA}^{\top}=\mathrm{A}^{\top} \mathrm{A}=\mathrm{I} \text {. Also } \mathrm{A}^{\top}=\mathrm{A}, \mathrm{so}^2=\mathrm{I} \Rightarrow \mathrm{A} \text { is an involuntary matrix. } \\
& \quad \Rightarrow\left|\mathrm{A}^2\right|=|\mathrm{A}|^2=1 \text { or }|\mathrm{A}|= \pm 1 \\
& \left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|=(a+b+c)\left|\begin{array}{lll}
1 & b & c \\
1 & c & a \\
1 & a & b
\end{array}\right|=(a+b+c)\left(a b+b c+c a-a^2-b^2-c^2\right)
\end{aligned}
$

A| = ab + bc + ca – a² – b² – c²

a² + b² + c² – ab – bc – ca 0

So |A| = -1. Hence a³ + b³ + c³ – 3abc = 1.

Again a² + b² + c² – ab – bc – ca = 1 1 – 3(ab + bc + c(A) = 1, so ab + bc + ca = 0

At least one of a, b, and c is negative.

Hence, the answer is the option (4).

Example 5: Let $A=\left[\begin{array}{ccc}x & y & z \\ y & z & x \\ z & x & y\end{array}\right]$, where $\mathrm{x}, \mathrm{y}$, and z are real numbers such that $x+y+z>0$ and $x y z=2$.If $A^2=I_3$, then the value of $x^3+y^3+z^3$ is
[JEE MAINS
[2021]
1) 7
2) 2
3) 5
4) 9

Solution

$
\begin{aligned}
& \mathrm{A}^2=\mathrm{I} \\
\Rightarrow & \mathrm{AA}^{\prime}=\mathrm{I}\left(\text { as } \mathrm{A}^{\prime}=\mathrm{A}\right)
\end{aligned}
$

$\Rightarrow \mathrm{A}$ is orthogonal

$
\begin{aligned}
& \text { So, } x^2+y^2+z^2=1 \text { and } x y+y z+z x=0 \\
& \Rightarrow(x+y+z)^2=1+2 \times 0 \\
& \Rightarrow x+y+z=1 \\
& a^3+b^3+c^3=(a+b+c)\left[\left(a^2+b^2+c^2\right)-(a b+b c+c a)\right]+3 a b c
\end{aligned}
$

Thus,

$
\mathrm{x}^3+\mathrm{y}^3+\mathrm{z}^3=3 \times 2+1 \times(1-0)=7
$

Hence, the answer is the option 1.

Hence, the answer is the option 1.

Frequently Asked Questions (FAQs)

Q1) What is an orthogonal matrix?

Answer: A matrix is said to be an orthogonal matrix if the product of a matrix and its transpose gives an identity matrix. A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

Q2) If A and B are orthogonal then AB is orthogonal or not?

Answer: The product of two orthogonal matrices is also an orthogonal matrix. If A and B are orthogonal then AB is also orthogonal.

Q3) What is the determinant of orthogonal matrices?

Answer: The determinant of an orthogonal matrix is always equal to the -1 or +1. If A is orthogonal then | A| =1 or -1

Q4) Are all square matrices are orthogonal matrices?

Answer: No, All orthogonal matrices are square matrices but not all square matrices are orthogonal.

Q5) What are square matrices?

Answer: The square matrix is the matrix in which the number of rows = number of columns. So a matrix$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] \mathrm{m} \times \mathrm{n}$ is said to be a square matrix when m = n