91Ó°ÊÓ

Prove that the scalar 1 is the identity for scalar multiplication: \(1 A=A\)

Determine whether the matrix is idempotent. A square matrix \(A\) is idempotent if \(A^{2}=A\) $$\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$$

Prove the following distributive property: \((c+d) A=c A+d A\)

find the trace of the matrix. The trace of an \(n \times n\) matrix \(A\) is the sum of the main diagonal entries. That is, \(\operatorname{Tr}(A)=a_{11}+a_{22}+\cdots+a_{n n}\) $$\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$

Prove that if \(A^{2}=A,\) then \(I-2 A=(I-2 A)^{-1}\)

Determine whether the matrix is idempotent. A square matrix \(A\) is idempotent if \(A^{2}=A\) $$\left[\begin{array}{rr} 2 & 3 \\ -1 & -2 \end{array}\right]$$

find the trace of the matrix. The trace of an \(n \times n\) matrix \(A\) is the sum of the main diagonal entries. That is, \(\operatorname{Tr}(A)=a_{11}+a_{22}+\cdots+a_{n n}\) $$\left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & 1 & -1 & 2 \\ 4 & 2 & 1 & 0 \\ 0 & 0 & 5 & 1 \end{array}\right]$$

Determine whether the matrix is idempotent. A square matrix \(A\) is idempotent if \(A^{2}=A\) $$\left[\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}\right]$$

find the trace of the matrix. The trace of an \(n \times n\) matrix \(A\) is the sum of the main diagonal entries. That is, \(\operatorname{Tr}(A)=a_{11}+a_{22}+\cdots+a_{n n}\) $$\left[\begin{array}{rrrr} 1 & 4 & 3 & 2 \\ 4 & 0 & 6 & 1 \\ 3 & 6 & 2 & 1 \\ 2 & 1 & 1 & -3 \end{array}\right]$$

Complete the proof of Theorem 2.3 (a) Prove the associative property of multiplication: \(A(B C)=(A B) C\) (b) Prove the distributive property: \((A+B) C=A C+B C\) (c) Prove the property: \(c(A B)=(c A) B=A(c B)\)