Solving System of Linear Equations Using Inverse

Q. Let $A^{-1}$ be a $3\times 3$ matrix such that $A^2-5A+7I=0.$
Statement-I :-$A^{-1}=\frac{1}{7}(5I-A).$
Statement-II :-The polynomial $A^3-2A^2-3A+I$ can be reduced to $5(A-4I)$.Then

1. Both the statement are true.
2. Statement-I is true, but Statement-II is false.
3. Statement-I is false, but Statement-II is true.
4. Both the statement are false.

Q.

Which of the following options holds true for the system of equations,

x + y + z =6

x + 2y + 3z = 12

x + 4y + 7z =30

1. No solution

2. Infinite solution

3. Trivial solution only

4. Unique Non trivial solutions

Q. By the method of matrix inversion, solve the system.
$\left(\begin{array}{ccc}1& 1& 1\\ 2& 5& 7\\ 2& 1& -1\end{array}\right]\left(\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\\ x_3& y_3\end{array}\right]=\left(\begin{array}{cc}9& 2\\ 52& 15\\ 0& -1\end{array}\right]$

1. $x_1=-1,x_2=-3,x_3=5y_1=1,y_2=2,y_3=1$
2. $x_1=2,x_2=3,x_3=5y_1=-3,y_2=2,y_3=1$
3. $x_1=1,x_2=3,x_3=5y_1=-1,y_2=2,y_3=1$
4. $x_1=2,x_2=-3,x_3=-5y_1=1,y_2=2,y_3=1$

Q.

Number of solutions for the system of linear equations, will be,

x + y + z =6

x + 2y + 3z = 14

x + 4y + 2z =30

1. No solution

2. Infinite solution

3. Trivial solution only

4. Unique Non trivial solutions

Q.

For a system of n linear equations in n variables, let A, B and X represent the coefficient, constant and variable matrices.If $\left|A\right|=0$, and (adj A). B $\ne$ 0 the system of equations is inconsistent and has no solution.

1. True

2. False

Q.

For a system of n non homogeneous equations in n variables equations let A, B and X represent the coefficient, constant and variable matrices. Then,

If $\left|A\right|\ne$ 0, then X = $A^{-1}$ B gives a trivial solution.

1. True

2. False

Q.

Find the principal and general solutions of the equation

Q.

The equation x + 2y + 5z = 0 has a trivial solution

1. True

2. False