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12th Maths Weekly Test-2 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test-2 (Applications Of Matrices And Determinants)-Aug 2020

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

3
4
2
5

If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =

\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] \)
\(\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right] \)
\(\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right] \)

If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

15
12
14
11

If A, B and C are invertible matrices of some order, then which one of the following is not true?

adj A = |A|A^-1
adj(AB) = (adj A)(adj B)
det A^-1 = (det A)^-1
(ABC)^-1 = C^-1B^-1A^-1

If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I , then B =

\(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) A\)
\(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) { A }^{ T }\)
\(\left( \cos ^{ 2 }{ \theta } \right) I\)
(Sin²\(\frac { \theta }{ 2 } \))A

The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

1
2
4
3

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A|I

Only (i)
(ii) and (iii)
(iii) and (iv)
(i), (ii) and (iv)

If ρ(A) = ρ([A | B]), then the system AX = B of linear equations is

consistent and has a unique solution
consistent
consistent and has infinitely many solution
inconsistent

The augmented matrix of a system of linear equations is \(\left[ \begin{matrix} 1 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 1 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 7 \\ \begin{matrix} 4 \\ \lambda -7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ \mu +5 \end{matrix} \end{matrix} \right] \). The system has infinitely many solutions if

λ = 7, μ ≠ -5
λ = 7, μ = 5
λ ≠ 7, μ ≠ -5
λ = 7, μ = -5

If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is

\(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right] \)
\(\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right] \)

If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then

a² + b² + c² = 1
abc ≠ 1
a + b + c =0
a² + b² + c² + 2abc =1

Let A be a 3 x 3 matrix and B its adjoint matrix If |B|=64, then |A|=

±2
±4
±8
±12

The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is

0
1
2
infinitely many

If A is a square matrix of order n, then |adj A| =

|A|^n-1
|A|^n-2
|A|ⁿ
None

If A is a matrix of order m x n, then \(\rho\)(A) is

m
n
≤ min (m,n)
≥ min (m,n)

If \(\rho\)(A) = \(\rho\)([A/B]) = number of unknowns, then the system is

consistent and has infinitely many solutions
consistent
inconsistent
consistent and has unique solution

Every homogeneous system ______

Is always consistent
Has only trivial solution
Has infinitely many solution
Need not be consistent

In a homogeneous system if \(\rho\) (A) =\(\rho\)([A|0]) < the number of unknouns then the system has ________

trivial solution
only non - trivial solution
no solution
trivial solution and infinitely many non - trivial solutions

If A = [2 0 1] then the rank of AA^T is ______

1
2
3
0

In a square matrix the minor M_ij and the' co-factor A_ij of and element a_ij are related by _____

A_ij = -M_ij
A_ij = M_ij
A_ij = (-1)^i+j M_ij
A_ij =(-1)^i-j M_ij

Part - B

Generic 5 - Answer All Question

Using determinants; find the quadratic defined by f(x) =ax² + bx + c, if f(1) =0, f(2) =-2 and f(3) = -6.

For what value of λ, the system of equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ² is consistent.

If A = \(\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & 4 \\ 2 & -4 & 3 \end{matrix} \right] \), verify thatA(adj A)=(adj A)A = |A| I₃.

If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.

Find the inverse of A = \(\left[ \begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2 \end{matrix} \right] \) by Gauss-Jordan method.

Test for consistency of the following system of linear equations and if possible solve:
x + 2y - z = 3, 3x - y + 2z = 1, x - 2y + 3z = 3, x - y + z + 1 = 0

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