Express the matrix A =\(\begin{bmatrix} 1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5 \end{bmatrix}\)as the sum of a symmetric and a skew-symmetric matrices.

If A =\(\begin{bmatrix} 1 &0 &2 \\0 & 2 & 1 \\2 &0 &3 \end{bmatrix}\) and A³ - 6A² + 7A + KI = O, find the value of k.

Show that f(x) f(y) = f(x + y), where f(x) =\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

Compute all minors, cofactors of A and hence compute |A| if A =\(\begin{bmatrix} 1& 3 &-2 \\4 & -5 &6 \\ -3 & 5 & 2 \end{bmatrix}\) .
Also check that | A | remains unaltered by expanding along any row or any column.

Without expanding the determinants, show that | B | = 2| A |.
Where B =\(\begin{bmatrix} b+c & c+a & a+b \\ c+a & a+b &b+c \\a+b & b+c & c+a \end{bmatrix}\)and A =\(\begin{bmatrix} a& b & c \\ b & c & a \\ c & a & b \end{bmatrix}\)

Prove that \(\begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \end{vmatrix}=4a^2b^2c^2\)

If \(\begin{vmatrix} a & b &a\alpha +b \\ b & c & b\alpha+c\\ a\alpha+b & b \alpha+c &0 \end{vmatrix}=0.\) 
prove that a, b, c are in G.P. or \(\alpha\) is a root of ax² + 2bx + c = 0.

Prove that \(\begin{vmatrix} 1& a & a^2-bc \\1 &b &b^2-ca \\ 1 & c & c^2-ab \end{vmatrix}=0.\)

If a, b, c are p^th, q^th and r^th terms of an A.P, find the value of \(\begin{vmatrix} a & b & c \\ p & q & r \\ 1& 1 &1 \end{vmatrix}\)

Show that \(\begin{vmatrix} a^2+x^2& ab & ac \\ ab & b^2+x^2 & bc \\ ac &bc &c^2+x^2 \end{vmatrix}\) is divisible by x⁴.