The value of \(\begin{vmatrix} 2x+y & x & y \\ 2y+z & y & z \\ 2z+x & z & x \end{vmatrix}\) is ________.

The inverse matrix of \(\begin{pmatrix} \frac { 1 }{ 5 } & \frac { 5 }{ 25 } \\ \frac { 2 }{ 5 } & \frac { 1 }{ 2 } \end{pmatrix}\) is ________.

The value of \(\begin{vmatrix} 5 & 5 & 5 \\ 4x & 4y & 4z \\ -3x & -3y & -3z \end{vmatrix}\)is ________.

If A is 3 \(\times\) 3 matrix and |A| = 4, then |A^-1| is equal to ________.

If any three rows or columns of a determinant are identical then the value of the determinant is ________.

The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.50 & 0.30 \\ 0.41 & 0.33 \end{bmatrix}\). Test whether the system is viable as per Hawkins Simon conditions.

Solve: \(\begin{vmatrix}2& x&3\\4&1&6\\1&2&7 \end{vmatrix}=0\)

If A \(=\begin{bmatrix} 1 \\ -4\\3 \end{bmatrix}\) and  B = [-1 2 1], verify that (AB)^T = B^T. A^T

Find the values of x if \(\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.\)

Using the property of determinants show that \(\begin{vmatrix} x &a &x+a \\ y & b &y+b \\z & c & z+c \end{vmatrix}=0.\)

Evaluate \(\begin{vmatrix}10041 & 10042 & 10043 \\10045 & 10046 & 10047\\ 10049 & 10050 & 10051 \end{vmatrix}\)

Verify that A(adj A) = (adj A) A = IAI·I for the matrix A = \(\begin{bmatrix}2 & 3 \\-1 & 4\end{bmatrix}\)

Show that \(\begin{vmatrix} a & a+b&a+b+c \\2a &3a+2b &4a+3b+2c\\3a&6a+3b&10a+6b+3c \end{vmatrix}=a^3.\)

Find the numbers a and b such that A² + aA + bI = 0 for the matrix A =\(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\)

The technology matrix of an economic system of two industries is \(\left[ \begin{matrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{matrix} \right] \) Test whether the system is viable as per Hawkins – Simon conditions.

Evaluate:\(\begin{vmatrix} 1&a&a^2-bc\\1&b&b^2-ca\\1&c&c^2-ab \end{vmatrix}\)

The data are about an economy of two industries A and B. The values are in crores of rupees.

Find the output when the final demand changes to 300 for A and 600 for B.

Show that the matrix A =\(\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \)satisfies the equation A² - 4A - 5I₃ = 0 and hence find A^-1.

Weekly expenditure in an office for three weeks is given as follows. Assuming that the salary in all the three weeks of different categories of staff did not vary, calculate the salary for each type of staff, using matrix inversion method.