Understanding the Properties of matrix A

The question asks us to classify the given matrix \(A = \left[ {\begin{array}{*{20}{c}} 1&1&3\\ 5&2&6\\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]\) based on specific matrix properties: Nilpotent, Idempotent, and Scalar. We need to check if this matrix A fits any of these definitions.

Defining Different Matrix Types

Let's quickly define the matrix types mentioned in the options:

• Nilpotent Matrix: A square matrix \(M\) is called Nilpotent if \(M^k = 0\) for some positive integer \(k\), where 0 represents the zero matrix of the same size.
• Idempotent Matrix: A square matrix \(M\) is called Idempotent if \(M^2 = M\).
• Scalar Matrix: A Scalar matrix is a diagonal matrix where all the elements on the main diagonal are equal. It can be written as \(kI\), where \(k\) is a scalar value and \(I\) is the identity matrix.

Checking if matrix A is a Scalar Matrix

The given matrix A is: \(A = \left[ {\begin{array}{*{20}{c}} 1&1&3\\ 5&2&6\\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]\) A Scalar matrix must be a diagonal matrix, meaning all its non-diagonal elements must be zero. Looking at matrix A, we see elements like \(A_{12} = 1\), \(A_{21} = 5\), etc., which are not zero. Therefore, matrix A is not a Scalar matrix.

Calculating A² to Check for Idempotent or Nilpotent Properties

To check if matrix A is Idempotent or Nilpotent (specifically if \(A^2 = 0\)), we must calculate the product of matrix A with itself, i.e., \(A^2 = A \times A\).

Let's perform the matrix multiplication:

\(A^2 = \left[ {\begin{array}{*{20}{c}} 1&1&3\\ 5&2&6\\ { - 2}&{ - 1}&{ - 3} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 1&1&3\\ 5&2&6\\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]\)

We calculate each element of the resulting \(3 \times 3\) matrix:

• Element (1,1): \((1 \times 1) + (1 \times 5) + (3 \times -2) = 1 + 5 - 6 = 0\)
• Element (1,2): \((1 \times 1) + (1 \times 2) + (3 \times -1) = 1 + 2 - 3 = 0\)
• Element (1,3): \((1 \times 3) + (1 \times 6) + (3 \times -3) = 3 + 6 - 9 = 0\)
• Element (2,1): \((5 \times 1) + (2 \times 5) + (6 \times -2) = 5 + 10 - 12 = 3\)
• Element (2,2): \((5 \times 1) + (2 \times 2) + (6 \times -1) = 5 + 4 - 6 = 3\)
• Element (2,3): \((5 \times 3) + (2 \times 6) + (6 \times -3) = 15 + 12 - 18 = 9\)
• Element (3,1): \( permeates (-2 \times 1) + (-1 \times 5) + (-3 \times -2) = -2 - 5 + 6 = -1\)
• Element (3,2): \( permeates (-2 \times 1) + (-1 \times 2) + (-3 \times -1) = -2 - 2 + 3 = -1\)
• Element (3,3): \( permeates (-2 \times 3) + (-1 \times 6) + (-3 \times -3) = -6 - 6 + 9 = -3\)

So, the matrix \(A^2\) is:

Analyzing the Result for matrix A

We found that \(A^2 = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 3&3&9\\ { - 1}&{ - 1}&{ - 3} \end{array}} \right]\). This is not the zero matrix, so matrix A is not Nilpotent with an index of 2. We also compare \(A^2\) with the original matrix A. Since \(A^2 \neq A\), matrix A is not Idempotent.

Based on our analysis, the given matrix A does not satisfy the conditions for being a Nilpotent matrix (with index 2), an Idempotent matrix, or a Scalar matrix.

Conclusion

Since matrix A does not fit the definitions of Nilpotent, Idempotent, or Scalar matrices, the correct classification among the given options is "None of these".