If \(A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]\), then which of the following statements are correct? 1. An will always be singular for any positive integer n. 2. An will always be a diagonal matrix for any positive integer n. 3. An will always be a symmetric matrix for any positive integer n. Select the correct answer using the code given below:
2 and 3 only
Analyzing Matrix Properties for An
The question asks us to examine the properties of \(A^n\), where \(A\) is a given matrix and \(n\) is a positive integer. The matrix \(A\) is given as:
\(A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]\)
This is a special type of matrix known as a diagonal matrix because all the non-diagonal elements are zero.
Understanding An for a Diagonal Matrix
When we raise a diagonal matrix to a positive integer power \(n\), we simply raise each diagonal element to that power. For the given matrix \(A\), \(A^n\) is:
\(A^n = \left[\begin{array}{lll} 2^n & 0 & 0 \\ 0 & 3^n & 0 \\ 0 & 0 & 4^n \end{array}\right]\)
Now let's evaluate each statement provided in the question.
Statement 1: An will always be singular for any positive integer n.
A matrix is singular if its determinant is equal to zero. Let's find the determinant of \(A^n\).
The determinant of a diagonal matrix is the product of its diagonal elements.
\(\det(A^n) = 2^n \times 3^n \times 4^n\)
We can also express this as:
\(\det(A^n) = (2 \times 3 \times 4)^n = 24^n\)
For any positive integer \(n\), \(24^n\) will always be a positive number, and thus it will never be equal to zero.
Since the determinant of \(A^n\) is never zero, \(A^n\) is always non-singular (or invertible) for any positive integer \(n\).
Therefore, Statement 1 is incorrect.
Statement 2: An will always be a diagonal matrix for any positive integer n.
A diagonal matrix is a square matrix where all elements outside the main diagonal are zero. The elements on the main diagonal can be any value.
We found that \(A^n = \left[\begin{array}{lll} 2^n & 0 & 0 \\ 0 & 3^n & 0 \\ 0 & 0 & 4^n \end{array}\right]\).
In this matrix, the elements at positions \((1,2), (1,3), (2,1), (2,3), (3,1), (3,2)\) are all zero. The elements on the main diagonal are \(2^n, 3^n, 4^n\).
By the definition of a diagonal matrix, \(A^n\) is indeed a diagonal matrix for any positive integer \(n\).
Therefore, Statement 2 is correct.
Statement 3: An will always be a symmetric matrix for any positive integer n.
A square matrix \(M\) is called symmetric if it is equal to its transpose, i.e., \(M = M^T\). The transpose of a matrix is obtained by swapping its rows and columns.
Let's find the transpose of \(A^n\):
\((A^n)^T = \left[\begin{array}{lll} 2^n & 0 & 0 \\ 0 & 3^n & 0 \\ 0 & 0 & 4^n \end{array}\right]^T\)
Swapping rows and columns:
\((A^n)^T = \left[\begin{array}{lll} 2^n & 0 & 0 \\ 0 & 3^n & 0 \\ 0 & 0 & 4^n \end{array}\right]\)
We can see that \((A^n)^T\) is exactly the same as \(A^n\).
Since \(A^n = (A^n)^T\), \(A^n\) is always a symmetric matrix for any positive integer \(n\).
Therefore, Statement 3 is correct.
Summary of Statement Analysis
- Statement 1: \(A^n\) is always singular. (Incorrect)
- Statement 2: \(A^n\) is always a diagonal matrix. (Correct)
- Statement 3: \(A^n\) is always a symmetric matrix. (Correct)
The correct statements are 2 and 3.
Conclusion
Based on the analysis of each statement, the statements that are correct are 2 and 3.
| Statement | Analysis Result |
|---|---|
| 1. \(A^n\) is always singular | Incorrect (Determinant is \(24^n \ne 0\)) |
| 2. \(A^n\) is always a diagonal matrix | Correct (Structure remains diagonal) |
| 3. \(A^n\) is always a symmetric matrix | Correct (Transpose is the same) |
Revision Table: Matrix Properties
| Matrix Type / Property | Definition | Check for \(A^n\) |
|---|---|---|
| Singular Matrix | Determinant is 0 | \(\det(A^n) = 24^n \ne 0\). Not singular. |
| Diagonal Matrix | Non-diagonal elements are zero | \(A^n = \left[\begin{array}{lll} 2^n & 0 & 0 \\ 0 & 3^n & 0 \\ 0 & 0 & 4^n \end{array}\right]\). Is diagonal. |
| Symmetric Matrix | Matrix is equal to its transpose (\(M = M^T\)) | \((A^n)^T = A^n\). Is symmetric. |
Additional Information: Properties of Diagonal and Symmetric Matrices
Diagonal Matrices:
- Multiplying two diagonal matrices results in another diagonal matrix.
- The inverse of a diagonal matrix (if it exists, i.e., no diagonal elements are zero) is found by taking the reciprocal of each diagonal element.
- Raising a diagonal matrix to a power involves raising each diagonal element to that power.
- The determinant is the product of the diagonal elements.
- The eigenvalues of a diagonal matrix are its diagonal elements.
Symmetric Matrices:
- A matrix \(M\) is symmetric if \(M_{ij} = M_{ji}\) for all indices \(i\) and \(j\).
- Diagonal matrices are a special case of symmetric matrices because \(D_{ij} = 0\) for \(i \ne j\), and swapping indices still results in 0.
- The eigenvalues of a real symmetric matrix are always real numbers.
- Symmetric matrices are always diagonalizable by an orthogonal matrix.
Understanding these basic matrix properties helps in solving problems involving matrix operations and powers.
Important Questions from Operations on Matrices
If \(A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\), then what is the value of det(I + AA'), where I is the 3 × 3 identity matrix?
If \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\), then what is 23A3 - 19A2 - 4A equal to ?
If A is an orthogonal matrix of order 3 and \({\rm{B}} = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ { - 3}&0&2\\ 2&5&0 \end{array}} \right]\) , then which of the following is/are correct?
1. |AB| = ± 47
2. AB = BA
Select the correct answer using the code given below:If \({\rm{E}}\left( {\rm{\theta }} \right) = \left[ {\begin{array}{*{20}{c}} {\cos {\rm{\theta }}}&{\sin {\rm{\theta }}}\\ { - \sin {\rm{\theta \;}}}&{\cos {\rm{\theta }}} \end{array}} \right]\) then E(α) E(β) is equal to
If \({\rm{X}} = \left[ {\begin{array}{c} 3&{ - 4}\\ 1&{ - 1} \end{array}} \right],{\rm{\;B}} = \left[ {\begin{array}{c} 5&2\\ { - 2}&1 \end{array}} \right]{\rm{\;and\;A}} = \left[ {\begin{array}{c} {\rm{p}}&{\rm{q}}\\ {\rm{r}}&{\rm{s}} \end{array}} \right]\)
Satisfy the equation AX = B, then the matrix A is equal to
