11th Business Maths Monthly Test - 2 ( Matrices and Determinants )

MABS Institution

11th Business Maths Monthly Test - 2 ( Matrices and Determinants )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The Inverse of matrix of\(\begin{pmatrix} 3 & 1 \\ 5 & 2\end{pmatrix}\) is

\(\begin{pmatrix} 2 & -1 \\-5 & 3 \end{pmatrix}\)
\(\begin{pmatrix} -2 & 5 \\1 & -3 \end{pmatrix}\)
\(\begin{pmatrix} 3 & -1 \\-5 & -3 \end{pmatrix}\)
\(\begin{pmatrix} -3 & 5 \\1 & -2 \end{pmatrix}\)

If A is a square matrix of order 3, then |kA| is

k|A|
-k|A|
k³|A|
-k³|A|

If A is an invertible matrix of order 2, then det (A^-1) be equal to

det (A)
\({{1}\over{det(A)}}\)
1
0

The value of x if \(\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix}=0\) is

0, -1
0, 1
-1, 1
-1, -1

Which of the following matrix has no inverse

\(\begin{pmatrix} -1 & 1 \\ 1 &-4 \end{pmatrix}\)
\(\begin{pmatrix} 2 & -1 \\ -4 &2 \end{pmatrix}\)
\(\begin{pmatrix} cos\ a & sin\ a \\ -sin\ a & cos\ a \end{pmatrix}\)
\(\begin{pmatrix} sin\ a & cos\ a \\ -cos\ a & sin\ a \end{pmatrix}\)

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

\({{7}\over{30}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { 5 }{ 12 } \\ \frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)
\({{7}\over{30}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { -5 }{ 12 } \\ \frac { -2 }{ 5 } & \frac { 1 }{ 5 } \end{pmatrix}\)
\({{30}\over{7}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { 5 }{ 12 } \\ \frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)
\({{30}\over{7}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { -5 }{ 12 } \\ \frac { -2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)

If \(\begin{vmatrix} x & 2 \\ 8 &5 \end{vmatrix}=0\) then the value of x is

\({{-5}\over{6}}\)
\({{5}\over{6}}\)
\({{-16}\over{5}}\)
\({{16}\over{5}}\)

If any three-rows or columns of a determinant are identical, then the value of the determinant is

adj (AB) is equal to

adj A adj B
adj A^T adj B^T
adj B adj A
adj B^T adj A^T

If A is a square matrix of order 3 and IAI = 3 then | adj A| is equal to

Part - B

Generic 2 - Answer All Question

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

The technology matrix of an economic system of two industries is \(\begin{bmatrix} 0.50 & 0.25 \\ 0.40 & 0.67 \end{bmatrix}\). Test whether the system is viable as per Hawkins-Simon conditions.

Evaluate: \(\begin{bmatrix} 3&-2&4\\2&0&1\\1&2&3 \end{bmatrix}\)

Evaluate\(\left| \begin{matrix} 1 & 3 & 4 \\ 102 & 18 & 36 \\ 17 & 3 & 6 \end{matrix} \right| \)

Using the property of determinant, evaluate \(\begin{vmatrix} 6 &5 &12 \\ 2 & 4 &4 \\2 & 1 & 4 \end{vmatrix}.\)

Find the minors and cofactors of all the elements of the following determinants.
\(\begin{bmatrix} 1&-3&2\\4&-1&2\\3&5&2 \end{bmatrix}\)

The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.6 & 0.9 \\ 0.20 & 0.80 \end{bmatrix}\) .Test whether the system is viable as per Hawkins-Simon conditions.

Part - C

Generic 3 - Answer All Question

Find the adjoint of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ 0 & 5 & 1 \\ 3 & 6 & 8 \end{matrix} \right] \)

Find the adjoint of the matrix \(A=\begin{bmatrix}2&3\\1&4 \end{bmatrix}\)

If A=\(\left[ \begin{matrix} 2 & -2 & 2 \\ 2 & 3 & 0 \\ 9 & 1 & 5 \end{matrix} \right] \)then, show that (adj A)A=0.

Prove that \(\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right| \) is independent of \(\theta\)

The data below are about an economy of two industries P and Q. The values are in lakhs of rupees.

Producer User Final Demand Total output

P Q

P 16 12 12 40

Q 12 8 4 24

Find the technology matrix and check whether the system is viable as per Hawkins-Simon conditions.

Producer	User	Final Demand	Total output
P	16	12	12	40
Q	12	8	4	24

Find λ if the matrix \(\left[ \begin{matrix} 1 & 1 & 3 \\ 2 & \lambda & 4 \\ 9 & 7 & 11 \end{matrix} \right] \)has no inverse.

If A=\(\begin{bmatrix} 2 & 3 \\ 1 & -6 \end{bmatrix}\) and B=\(\begin{bmatrix} -1 & 4 \\ 1 & -2 \end{bmatrix}\) then verify adj(AB)=(adj B) (adj A).

Part - D

Generic 5 - Answer All Question

If a, b, c are in A.P, find the value of \(\left| \begin{matrix} 2y+4 & \quad 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{matrix} \right| \)

The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third we get 23. If we multiply the first by 3 and add second and third to it, we get 46. By using matrix inversion method find the numbers.

The following inter – industry transactions table was constructed for an economy of the year 2016

Industry 1 2 Final consumption Total output

1 500 1,600 400 2,500

2 1,750 1,600 4,650 8,000

Labours 250 4,800 - -

Construct technology co-efficient matrix showing direct requirements. Does a solution exist for this system.

Find the minor and cofactor of each element of the determinant.\(\left| \begin{matrix} 3 & 1 & 2 \\ 2 & 2 & 5 \\ 4 & 1 & 0 \end{matrix} \right| \)
If A=\(\left[ \begin{matrix} -1 & 2 & -2 \\ 4 & -3 & 4 \\ 4 & -4 & 5 \end{matrix} \right] \)then, show that the inverse of A is A itself.

Let a, b and c denote the sides BC, CA and AB repectively of \(\Delta\) ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0\), then find the value of sin² A+sin²B+sin²C.

Solve by matrix inversion method: 2x + 3y - 5 = 0, x - 2y + 1 = 0.

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Industry	1	2	Final consumption	Total output
1	500	1,600	400	2,500
2	1,750	1,600	4,650	8,000
Labours	250	4,800	-	-

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11th Business Maths Monthly Test - 2 ( Matrices and Determinants )

MABS Institution

11th Business Maths Monthly Test - 2 ( Matrices and Determinants )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

The Inverse of matrix of\(\begin{pmatrix} 3 & 1 \\ 5 & 2\end{pmatrix}\) is

If A is a square matrix of order 3, then |kA| is

If A is an invertible matrix of order 2, then det (A-1) be equal to

The value of x if \(\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix}=0\) is

Which of the following matrix has no inverse

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

If \(\begin{vmatrix} x & 2 \\ 8 &5 \end{vmatrix}=0\) then the value of x is

If any three-rows or columns of a determinant are identical, then the value of the determinant is

adj (AB) is equal to

If A is a square matrix of order 3 and IAI = 3 then | adj A| is equal to

Part - B

Generic 2 - Answer All Question

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

The technology matrix of an economic system of two industries is \(\begin{bmatrix} 0.50 & 0.25 \\ 0.40 & 0.67 \end{bmatrix}\). Test whether the system is viable as per Hawkins-Simon conditions.

Evaluate: \(\begin{bmatrix} 3&-2&4\\2&0&1\\1&2&3 \end{bmatrix}\)

Evaluate\(\left| \begin{matrix} 1 & 3 & 4 \\ 102 & 18 & 36 \\ 17 & 3 & 6 \end{matrix} \right| \)

Using the property of determinant, evaluate \(\begin{vmatrix} 6 &5 &12 \\ 2 & 4 &4 \\2 & 1 & 4 \end{vmatrix}.\)

Find the minors and cofactors of all the elements of the following determinants. \(\begin{bmatrix} 1&-3&2\\4&-1&2\\3&5&2 \end{bmatrix}\)

The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.6 & 0.9 \\ 0.20 & 0.80 \end{bmatrix}\) .Test whether the system is viable as per Hawkins-Simon conditions.

Part - C

Generic 3 - Answer All Question

Find the adjoint of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ 0 & 5 & 1 \\ 3 & 6 & 8 \end{matrix} \right] \)

Find the adjoint of the matrix \(A=\begin{bmatrix}2&3\\1&4 \end{bmatrix}\)

If A=\(\left[ \begin{matrix} 2 & -2 & 2 \\ 2 & 3 & 0 \\ 9 & 1 & 5 \end{matrix} \right] \)then, show that (adj A)A=0.

Prove that \(\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right| \) is independent of \(\theta\)

The data below are about an economy of two industries P and Q. The values are in lakhs of rupees. Producer User Final Demand Total output P Q P 16 12 12 40 Q 12 8 4 24 Find the technology matrix and check whether the system is viable as per Hawkins-Simon conditions.

Find λ if the matrix \(\left[ \begin{matrix} 1 & 1 & 3 \\ 2 & \lambda & 4 \\ 9 & 7 & 11 \end{matrix} \right] \)has no inverse.

If A=\(\begin{bmatrix} 2 & 3 \\ 1 & -6 \end{bmatrix}\) and B=\(\begin{bmatrix} -1 & 4 \\ 1 & -2 \end{bmatrix}\) then verify adj(AB)=(adj B) (adj A).

Part - D

Generic 5 - Answer All Question

If a, b, c are in A.P, find the value of \(\left| \begin{matrix} 2y+4 & \quad 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{matrix} \right| \)

The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third we get 23. If we multiply the first by 3 and add second and third to it, we get 46. By using matrix inversion method find the numbers.

Find the minor and cofactor of each element of the determinant.\(\left| \begin{matrix} 3 & 1 & 2 \\ 2 & 2 & 5 \\ 4 & 1 & 0 \end{matrix} \right| \)

If A=\(\left[ \begin{matrix} -1 & 2 & -2 \\ 4 & -3 & 4 \\ 4 & -4 & 5 \end{matrix} \right] \)then, show that the inverse of A is A itself.

Let a, b and c denote the sides BC, CA and AB repectively of \(\Delta\) ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0\), then find the value of sin2 A+sin2B+sin2C.

Solve by matrix inversion method: 2x + 3y - 5 = 0, x - 2y + 1 = 0.

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If A is an invertible matrix of order 2, then det (A^-1) be equal to

Find the minors and cofactors of all the elements of the following determinants.
\(\begin{bmatrix} 1&-3&2\\4&-1&2\\3&5&2 \end{bmatrix}\)

The data below are about an economy of two industries P and Q. The values are in lakhs of rupees.

Producer User Final Demand Total output

P Q

P 16 12 12 40

Q 12 8 4 24

Find the technology matrix and check whether the system is viable as per Hawkins-Simon conditions.

Let a, b and c denote the sides BC, CA and AB repectively of \(\Delta\) ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0\), then find the value of sin² A+sin²B+sin²C.