91Ó°ÊÓ

Evaluate the determinants in Exercises 1 and 2 . If \(A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]\), then show that \(|2 \mathrm{~A}|=4|\mathrm{~A}|\)

Using the property of determinants and without expanding in Exercises 1 to 7 , prove that: $$ \left|\begin{array}{lll} 1 & b c & a(b+c) \\ 1 & c a & b(c+a) \\ 1 & a b & c(a+b) \end{array}\right|=0 $$

Solve the equation \(\left|\begin{array}{ccc}x+a & x & x \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0, a \neq 0\)

Using the property of determinants and without expanding in Exercises 1 to 7 , prove that: $$ \left|\begin{array}{ccc} 0 & a & -b \\ -a & 0 & -c \\ b & c & 0 \end{array}\right|=0 $$

Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11 . $$ \begin{array}{lll} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & 1 \end{array} $$

Solve system of linear equations, using matrix method, in Exercises 7 to 14. $$ \begin{array}{l} 5 x+2 y=3 \\ 3 x+2 y=5 \end{array} $$

Let \(\mathrm{A}\) be a square matrix of order \(3 \times 3\), then \(|k \mathrm{~A}|\) is equal to (A) \(k|\mathrm{~A}|\) (B) \(k^{2}|\mathrm{~A}|\) (C) \(k^{3}|\mathrm{~A}|\) (D) \(3 k|\mathrm{~A}|\)