Let's analyze the given matrix and the two statements about its properties. The matrix is \({\rm{A}} = \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right)\).

We need to verify the correctness of the following statements:

• Statement 1: \({\rm{A}}^2 = -{\rm{A}}\)
• Statement 2: \({\rm{A}}^3 = 4{\rm{A}}\)

Verifying Statement 1: \({\rm{A}}^2 = -{\rm{A}}\)

First, we calculate the square of the matrix A, which is \({\rm{A}}^2 = {\rm{A}} \times {\rm{A}}\).

$${\rm{A}}^2 = \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right) \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right)$$

To perform matrix multiplication, we multiply rows of the first matrix by columns of the second matrix.

$${\rm{A}}^2 = \left( {\begin{array}{*{20}{c}} {(-1)(-1) + (1)(1)}&{(-1)(1) + (1)(-1)}\\ {(1)(-1) + (-1)(1)}&{(1)(1) + (-1)(-1)} \end{array}} \right)$$

$${\rm{A}}^2 = \left( {\begin{array}{*{20}{c}} {1 + 1}&{-1 - 1}\\ {-1 - 1}&{1 + 1} \end{array}} \right)$$

$${\rm{A}}^2 = \left( {\begin{array}{*{20}{c}} {2}&{-2}\\ {-2}&{2} \end{array}} \right)$$

Next, we calculate \(-{\rm{A}}\).

$$-{\rm{A}} = -1 \times \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {(-1)(-1)}&{(-1)(1)}\\ {(-1)(1)}&{(-1)(-1)} \end{array}} \right)$$

$$-{\rm{A}} = \left( {\begin{array}{*{20}{c}} {1}&{-1}\\ {-1}&{1} \end{array}} \right)$$

Now we compare \({\rm{A}}^2\) and \(-{\rm{A}}\):

$$\left( {\begin{array}{*{20}{c}} {2}&{-2}\\ {-2}&{2} \end{array}} \right) \neq \left( {\begin{array}{*{20}{c}} {1}&{-1}\\ {-1}&{1} \end{array}} \right)$$

Since the elements of the matrices are not equal, Statement 1 (\({\rm{A}}^2 = -{\rm{A}}\)) is incorrect.

Verifying Statement 2: \({\rm{A}}^3 = 4{\rm{A}}\)

We need to calculate \({\rm{A}}^3\). We already have \({\rm{A}}^2\), so \({\rm{A}}^3 = {\rm{A}}^2 \times {\rm{A}}\).

$${\rm{A}}^3 = \left( {\begin{array}{*{20}{c}} {2}&{-2}\\ {-2}&{2} \end{array}} \right) \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right)$$

Multiply the rows of the first matrix by the columns of the second matrix:

$${\rm{A}}^3 = \left( {\begin{array}{*{20}{c}} {(2)(-1) + (-2)(1)}&{(2)(1) + (-2)(-1)}\\ {(-2)(-1) + (2)(1)}&{(-2)(1) + (2)(-1)} \end{array}} \right)$$

$${\rm{A}}^3 = \left( {\begin{array}{*{20}{c}} {-2 - 2}&{2 + 2}\\ {2 + 2}&{-2 - 2} \end{array}} \right)$$

$${\rm{A}}^3 = \left( {\begin{array}{*{20}{c}} {-4}&{4}\\ {4}&{-4} \end{array}} \right)$$

Next, we calculate \(4{\rm{A}}\).

$$4{\rm{A}} = 4 \times \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {(4)(-1)}&{(4)(1)}\\ {(4)(1)}&{(4)(-1)} \end{array}} \right)$$

$$4{\rm{A}} = \left( {\begin{array}{*{20}{c}} {-4}&{4}\\ {4}&{-4} \end{array}} \right)$$

Now we compare \({\rm{A}}^3\) and \(4{\rm{A}}\):

$$\left( {\begin{array}{*{20}{c}} {-4}&{4}\\ {4}&{-4} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {-4}&{4}\\ {4}&{-4} \end{array}} \right)$$

Since the elements of the matrices are equal, Statement 2 (\({\rm{A}}^3 = 4{\rm{A}}\)) is correct.

Summary of Statements

• Statement 1: \({\rm{A}}^2 = -{\rm{A}}\) is Incorrect.
• Statement 2: \({\rm{A}}^3 = 4{\rm{A}}\) is Correct.

Therefore, only Statement 2 is correct.

Conclusion on Matrix Properties

Based on our calculations, only the second statement regarding the matrix A is correct. The first statement does not hold true for the given matrix A.

Revision Table: Matrix Calculations

Reviewing the core steps involved in checking these matrix properties:

• Calculating the square of a matrix (\({\rm{A}}^2\)).
• Calculating the cube of a matrix (\({\rm{A}}^3\)).
• Scalar multiplication of a matrix (\(-{\rm{A}}\) and \(4{\rm{A}}\)).
• Comparing matrices for equality.

These steps are fundamental in understanding matrix operations and verifying matrix properties.

Additional Information: Matrix Algebra Concepts

Understanding matrix operations like multiplication and scalar multiplication is crucial in linear algebra. Matrix multiplication is not commutative in general (\({\rm{AB}} \neq {\rm{BA}}\)), unlike scalar multiplication. Powers of a matrix (\({\rm{A}}^n\)) are found by multiplying the matrix by itself \(n\) times. Special types of matrices, like idempotent matrices (\({\rm{A}}^2 = {\rm{A}}\)) or nilpotent matrices (\({\rm{A}}^k = 0\) for some positive integer \(k\)), have specific power properties. The given matrix A exhibits a specific relationship between its powers \({\rm{A}}^2\) and \({\rm{A}}^3\) and the original matrix A, which was verified through calculation.