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.
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