(1) Matrix addition and the opposite matrix

Given are the matrices
\( A=\begin{pmatrix} 6 & 1 \\ 7 & 5 \end{pmatrix}, \quad B=\begin{pmatrix} 6 & 8 \\ 1 & 2 \end{pmatrix}. \)
a) Calculate the matrix ( A+B ).
b) Determine the opposite matrix of matrix ( A ) and then calculate ( A+(-A) ). Explain the obtained result.

(2) Matrix subtraction and property verification

Given are the matrices
\( C=\begin{pmatrix} -4 & -3 \\ 1 & 2 \end{pmatrix}, \quad D=\begin{pmatrix} 0 & 2 \\ 1 & -2 \end{pmatrix}. \)
a) Calculate the matrix ( C-D ).
b) Check if the equality ( C-D=-(D-C) ) holds.

(3) Scalar multiplication and distributivity

Given is the matrix
\( E=\begin{pmatrix} 2 & 6 \\ 5 & 3 \end{pmatrix} \)
and scalars ( k=0 ) and ( m=3 ).
a) Calculate the matrix ( kE ).
b) Calculate ( (k+m)E ) and compare with ( kE+mE ).

(4) Matrix product and non-commutativity

Given are the matrices
\( F=\begin{pmatrix} 5 & 2 \\ 4 & 3 \end{pmatrix}, \quad G=\begin{pmatrix} 4 & 5 \\ 2 & 1 \end{pmatrix}. \)
a) Calculate the product ( FG ).
b) Calculate the product ( GF ).
c) Compare the results and conclude whether matrix multiplication is commutative in this case.

(5) Transposition and transposition properties

Given is the matrix
\( H=\begin{pmatrix} 3 & 6 & 0 \\ 5 & 4 & 7 \end{pmatrix}. \)
a) Determine the transposed matrix ( H^T ).
b) Calculate ( (H^T)^T ) and compare with matrix ( H ).

(6) Distributivity of matrix multiplication

Given are the matrices
\( A=\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}, \quad B=\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}, \)
\( C=\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}. \)
a) Calculate ( A(B+C) ).
b) Calculate ( AB+AC ) and compare the results.

(7) Identity matrix and powers

Given is the matrix
\( K=\begin{pmatrix} 2 & 6 \\ 8 & 7 \end{pmatrix} \)
and the identity matrix ( I ) of order ( n=2 ).
a) Calculate ( KI ) and ( IK ).
b) Calculate ( K^2 ) and compare with ( KK ).

(8) Linear combination of matrices

Given are the matrices
\( M=\begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix}, \quad N=\begin{pmatrix} 2 & 1 \\ 4 & 6 \end{pmatrix}. \)
For scalars ( a=3 ) and ( b=5 ):
a) Calculate ( aM-bN ).
b) Determine the matrix ( X ) such that ( X+aN=bM ) holds.

(9) Square of a matrix and difference of squares

Given is the matrix
\( L=\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}. \)
a) Calculate ( L^2 ).
b) Calculate ( (L-I)(L+I) ), where ( I ) is the identity matrix of order 2.

(10) Associativity of matrix multiplication

Given are the matrices
\( P=\begin{pmatrix} 4 & 2 \\ 3 & 1 \end{pmatrix}, \quad Q=\begin{pmatrix} 3 & 2 \\ 4 & 1 \end{pmatrix}, \quad R=\begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}. \)
a) Calculate ( (PQ)R ).
b) Calculate ( P(QR) ).
c) Compare the obtained matrices and conclude whether the associativity of multiplication holds in this example.