12th Maths Weekly Test -1 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (Applications Of Matrices And Determinants)-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If A is a 3 × 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =

If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I - A =

A^-1
\(\frac { { A }^{ -1 } }{ 2 } \)
3A^-1
2A^-1

If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is

0
-2
-3
-1

If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
\(\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right] \)

\(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \)

\(\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right] \)

\(\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right] \)
If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is
\(\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right] \)

\(\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right] \)

\(\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right] \)

\(\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right] \)

If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \) then adj (AB) is

0
sin θ
cos θ
1

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 system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if

λ = 8
λ = 8, μ ≠ 36
λ ≠ 8
none

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

0
1
2
infinitely many

Which of the following is not an elementary transformation?

R_i ↔️ R_j
R_i ⟶ 2R_i + R_j
C_j ⟶ C_j + C_i
R_i ⟶ R_i + C_j

If A is a non-singular matrix then IA^-1|= ______

\(\left| \frac { 1 }{ { A }^{ 2 } } \right| \)
\(\frac { 1 }{ |A^{ 2 }| } \)
\(\left| \frac { 1 }{ A } \right| \)
\(\frac { 1 }{ |A| } \)

Part - B

Generic 2 - Answer All Question

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.

Find the matrix A for which A\(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] =\left[ \begin{matrix} 14 & 7 \\ 7 & 7 \end{matrix} \right] \).
Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ 5 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} -1 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 4 \\ 11 \end{matrix} \end{matrix} \right] \)

Solve the following systems of linear equations by Gaussian elimination method:
2x − 2y + 3z = 2, x + 2y − z = 3,3x − y + 2z = 1

Part - C

Generic 3 - Answer All Question

Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

Verify that (A^-1)^T = (A^T)^-1 for A=\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1

Solve the following system of linear equations, using matrix inversion method:
5x + 2y = 3, 3x + 2y = 5.
Under what co.nditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?

Solve the following system:
x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

Part - D

Generic 5 - Answer All Question

Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \)=2

Show that the matrix \(\left[ \begin{matrix} 3 & 1 & 4 \\ 2 & 0 & -1 \\ 5 & 2 & 1 \end{matrix} \right] \) is non-singular and reduce it to the identity matrix by elementary row transformations.
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.

Solve, by Cramer’s rule, the system of equations
x₁ - x₂ = 3, 2x₁ + 3x₂ + 4x₃ = 17, x₂ + 2x₃ = 7.

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12th Maths Weekly Test -1 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (Applications Of Matrices And Determinants)-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

If A is a 3 × 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT =

If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I - A =

If A is a non-singular matrix such that A-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (AT)−1 =

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

If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \) then adj (AB) is

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

The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if

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

Which of the following is not an elementary transformation?

If A is a non-singular matrix then IA-1|= ______

Part - B

Generic 2 - Answer All Question

Find the inverse (if it exists) of the following: \(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.

Find the matrix A for which A\(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] =\left[ \begin{matrix} 14 & 7 \\ 7 & 7 \end{matrix} \right] \).

Find the inverse (if it exists) of the following: \(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Solve the following systems of linear equations by Gaussian elimination method: 2x − 2y + 3z = 2, x + 2y − z = 3,3x − y + 2z = 1

Part - C

Generic 3 - Answer All Question

Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

Verify that (A-1)T = (AT)-1 for A=\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)-1 = B-1A-1

Solve the following system of linear equations, using matrix inversion method: 5x + 2y = 3, 3x + 2y = 5.

Under what co.nditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?

Solve the following system: x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

Part - D

Generic 5 - Answer All Question

Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \)=2

Show that the matrix \(\left[ \begin{matrix} 3 & 1 & 4 \\ 2 & 0 & -1 \\ 5 & 2 & 1 \end{matrix} \right] \) is non-singular and reduce it to the identity matrix by elementary row transformations.

If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A2 + xA + yI2 = O2. Hence, find A-1.

Solve, by Cramer’s rule, the system of equations x1 - x2 = 3, 2x1 + 3x2 + 4x3 = 17, x2 + 2x3 = 7.

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If A is a 3 × 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =

If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

If A is a non-singular matrix then IA^-1|= ______

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Solve the following systems of linear equations by Gaussian elimination method:
2x − 2y + 3z = 2, x + 2y − z = 3,3x − y + 2z = 1

Verify that (A^-1)^T = (A^T)^-1 for A=\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1

Solve the following system of linear equations, using matrix inversion method:
5x + 2y = 3, 3x + 2y = 5.

Solve the following system:
x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.

Solve, by Cramer’s rule, the system of equations
x₁ - x₂ = 3, 2x₁ + 3x₂ + 4x₃ = 17, x₂ + 2x₃ = 7.