Let A = \(\left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right)\) and A = P + Q where P is symmetric matrix and Q is skew-symmetric matrix.
What is P equal to ?
Understanding Matrix Decomposition and Symmetric Matrices
The question asks us to find the symmetric part, denoted by P, of a given matrix A. We are told that matrix A can be expressed as the sum of a symmetric matrix P and a skew-symmetric matrix Q, i.e., A = P + Q.
Any square matrix A can be uniquely decomposed into the sum of a symmetric matrix and a skew-symmetric matrix. The formulas for P and Q are:
- Symmetric part P: \(P = \frac{1}{2}(A + A^T)\)
- Skew-symmetric part Q: \(Q = \frac{1}{2}(A - A^T)\)
Here, \(A^T\) represents the transpose of matrix A.
Finding the Symmetric Part P
The given matrix A is:
\(A = \left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right)\)
First, let's find the transpose of matrix A, \(A^T\). The transpose is obtained by swapping the rows and columns of A.
\(A^T = \left(\begin{array}{ccc}0 & \cos^2 \theta & \sin^2 \theta \\ \sin^2 \theta & 0 & \cos^2 \theta \\ \cos^2 \theta & \sin^2 \theta & 0\end{array}\right)\)
Next, we calculate the sum of A and \(A^T\):
\(A + A^T = \left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right) + \left(\begin{array}{ccc}0 & \cos^2 \theta & \sin^2 \theta \\ \sin^2 \theta & 0 & \cos^2 \theta \\ \cos^2 \theta & \sin^2 \theta & 0\end{array}\right)\)
\(A + A^T = \left(\begin{array}{ccc}0+0 & \sin^2 \theta + \cos^2 \theta & \cos^2 \theta + \sin^2 \theta \\ \cos^2 \theta + \sin^2 \theta & 0+0 & \sin^2 \theta + \cos^2 \theta \\ \sin^2 \theta + \cos^2 \theta & \cos^2 \theta + \sin^2 \theta & 0+0\end{array}\right)\)
Using the trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\), the sum becomes:
\(A + A^T = \left(\begin{array}{ccc}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right)\)
Finally, we calculate P using the formula \(P = \frac{1}{2}(A + A^T)\):
\(P = \frac{1}{2} \left(\begin{array}{ccc}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right) = \left(\begin{array}{ccc}\frac{1}{2}(0) & \frac{1}{2}(1) & \frac{1}{2}(1) \\ \frac{1}{2}(1) & \frac{1}{2}(0) & \frac{1}{2}(1) \\ \frac{1}{2}(1) & \frac{1}{2}(1) & \frac{1}{2}(0)\end{array}\right)\)
\(P = \left(\begin{array}{ccc}0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0\end{array}\right)\)
Verifying the Result
To verify that P is indeed a symmetric matrix, we check if \(P^T = P\). The transpose of P is:
\(P^T = \left(\begin{array}{ccc}0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0\end{array}\right)^T = \left(\begin{array}{ccc}0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0\end{array}\right)\)
Since \(P^T = P\), P is a symmetric matrix.
The calculated matrix for P is:
\(\left(\begin{array}{ccc}0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0\end{array}\right)\)
Comparing this with the given options, we find that it matches Option 1.
Revision Table: Key Matrix Definitions
| Term | Definition | Property |
|---|---|---|
| Matrix Transpose (\(A^T\)) | Matrix obtained by interchanging rows and columns of A. | \((A^T)^T = A\), \((A+B)^T = A^T+B^T\), \((kA)^T = kA^T\), \((AB)^T = B^T A^T\) |
| Symmetric Matrix (P) | A square matrix where \(P^T = P\). | Elements are symmetric about the main diagonal (\(p_{ij} = p_{ji}\)). |
| Skew-Symmetric Matrix (Q) | A square matrix where \(Q^T = -Q\). | Elements on the main diagonal are zero (\(q_{ii} = 0\)), and \(q_{ij} = -q_{ji}\). |
Additional Information: Properties of Matrix Decomposition
Any square matrix A can be uniquely written as the sum of a symmetric matrix P and a skew-symmetric matrix Q. This decomposition is unique because if A = P + Q where P is symmetric and Q is skew-symmetric, then:
\(A^T = (P+Q)^T = P^T + Q^T = P - Q\)
We now have two equations:
- A = P + Q
- \(A^T\) = P - Q
Adding these two equations gives:
\(A + A^T = (P+Q) + (P-Q) = 2P\)
So, \(P = \frac{1}{2}(A + A^T)\).
Subtracting the second equation from the first gives:
\(A - A^T = (P+Q) - (P-Q) = 2Q\)
So, \(Q = \frac{1}{2}(A - A^T)\).
These unique formulas demonstrate how the symmetric and skew-symmetric parts are derived and why the decomposition is unique for any square matrix.
Similar Questions
If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct?
If \(A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right],\) then the matrix A is a/an
How many distinct matrices exist with all four entries taken from (1, 2)?
What is the order of \(\left[ {{\rm{x\;\;y\;\;z}}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\\ {\rm{h}}&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{x}}\\ {\rm{y}}\\ {\rm{z}} \end{array}} \right]?\)
What is Q equal to ?
What is the minimum value of determinant of A ?
If matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {1 - {\rm{i}}}&{\rm{i}}\\ { - {\rm{i}}}&{1 - {\rm{i}}} \end{array}} \right]\) where \(\rm i = \sqrt {-1},\) then which one of the following is correct?
If X is a matrix of order 3 × 3, Y is a matrix of order 2 × 3 and Z is a matrix of order 3 × 2, then which of the following are correct?
1. (ZY)X is a square matrix having 9 entries.
2. Y(XZ) is a square matrix having 4 entries.
3. X(YZ) is not defined.
Select the correct answer using the code given below :
Which one of the following matrices is an elementary matrix?
The matrix is \(\left[ {\begin{array}{c} 0&{ - 4 + i}\\ {4 + i}&0 \end{array}} \right]\)
Important Questions from Types of Matrices
If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct?
If the matrix A = \(\left[ {\begin{array}{c} 0&1&{ - 1} \\ 4&{ - 3}&4 \\ 3&{ - 3}&4 \end{array}} \right]\) = B + C, where B is symmetric and C is a skew-symmetric matrix, find the matrix B
If \(A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right],\) then the matrix A is a/an
How many distinct matrices exist with all four entries taken from (1, 2)?
Let \(A = \left[ {\begin{array}{*{20}{c}} 1&1&3\\ 5&2&6\\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]\), then A is
