Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
\(\begin{bmatrix} 4 & -2 \\ 3& -5 \end{bmatrix}\)

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
\(\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)

Construct the matrix \(A=[a_{ij}]_{3\times 3}\), where \(a_{ij}=i-j.\) State whether A is symmetric or skew-symmetric.

A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins, and almonds.
Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds.
Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds.
Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds.
The cost of 50 gm of cashew nuts is Rs.50, 50 gm of raisins is Rs.10, and 50 gm of almonds is Rs.60. What is the cost of each gift pack?

If a, b, c and x are positive real numbers, then show that \(\begin{vmatrix} (a^x+a^{-x})^2 &(a^x-a^{-x})^2 &1 \\ (b^x+b^{-x})^2 & (b^x-b^{-x})^2 & 1 \\ (c^x+c^{-x})^2 & (c^x-c^{-x})^2 & 1 \end{vmatrix}\) is zero.

Determine the roots of the equation \(\begin{vmatrix} 1 &4 &2 0 \\ 1 & -2 & 5 \\ 1 &2x &5x^2 \end{vmatrix}=0\)

Using cofactors of elements of second row, evaluate | A |, where A = \(\begin{bmatrix} 5 & 3 &8 \\ 2 & 0 & 1 \\1 &2 &3 \end{bmatrix}\)

Show that \(\begin{vmatrix} 0 & c &b \\ c & 0 &a \\ b & a & 0 \end{vmatrix}^2=\begin{vmatrix} b^2+c^2 & ab & ac \\ ab & c^2+a^2 & bc \\ ab & bc & a^2+b^2 \end{vmatrix}\)

Prove that \(\begin{vmatrix} 1 &x &x \\ x & 1 &x \\ x &x &1 \end{vmatrix}^2=\begin{vmatrix}1-2x^2 & -x^2 &-x^2 \\ -x^2 &-1 &x^2-2x \\ -x^2 &x^2-2x &-1 \end{vmatrix}\)

Find the value of the product \(\begin{vmatrix} log_364 &log_43 \\ log_38 & log_49 \end{vmatrix}\times \begin{vmatrix} log_23 & log_83 \\ log_34 & log_34 \end{vmatrix}\)