11th Business Maths Weekly Test -1 ( Matrices and Determinants )

MABS Institution

11th Business Maths Weekly Test -1 ( Matrices and Determinants )-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If A and B are non-singular matrices then, which of the following is incorrect?

A² = Iimplies A^-1= A
I^-1 = I
If AX = B, then X = B^-1 A
If A is square matrix of order 3 then |adj A|= |A|²

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 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 A \(=\begin{pmatrix} -1 & 2 \\ 1 & -4 \end{pmatrix}\) then A (adj A) is

\(\begin{pmatrix} -4 & -2 \\ -1 & -1 \end{pmatrix}\)
\(\begin{pmatrix} 4 & -2 \\ -1 & 1 \end{pmatrix}\)
\(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)
\(\begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}\)

The value of \(\begin{vmatrix} x & x^2 & -yz & 1 \\ y & y^2 & -zx & 1 \\ z & z^2 & -xy &1 \end{vmatrix}\) is

1
0
-1
-xyz

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

xyz
x+y+z
2x+2y+2z
0

The inventor of input-output analysis is

Sir Francis Galton
Fisher
Prof. Wassily W. Leontief
Arthur Caylay

Which of the following matrix has no inverse

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

If A is 3 x 3 matrix and |A|= 4, then |A^-1| is equal to

\({{1}\over{4}}\)
\({{1}\over{16}}\)
2
4

If A = \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)such that ad - bc \(\neq\) 0 then A^-1 is

\({{1}\over{ad-bc}}\begin{pmatrix} d & b \\-c & a\end{pmatrix}\)
\({{1}\over{ad-bc}}\begin{pmatrix} d & b \\c & a\end{pmatrix}\)
\({{1}\over{ad-bc}}\begin{pmatrix} d & -b \\-c & a\end{pmatrix}\)
\({{1}\over{ad-bc}}\begin{pmatrix} d & -b \\c & a\end{pmatrix}\)

Part - B

Generic 2 - Answer All Question

Show that \(\left[ \begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix} \right] \)is a singular matrix.

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

Find the minors and cofactors of all the elements of the following determinants.
\(\begin{bmatrix} 1&-3&2\\4&-1&2\\3&5&2 \end{bmatrix}\)
If \(A=\left| \begin{matrix} -2 & 6 \\ 3 & -9 \end{matrix} \right| \)then, find A^-1

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

Solve: 2x+ 5y = 1 and 3x + 2y = 7 using matrix method.

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

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).

Using the properties of determinants, show that \(\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| \)=0
Without actual expansion show that the value of the determinant \(\begin{vmatrix}5 &5^2 &5^3 \\5^2 & 5^3 & 5^4\\5^4&5^5&5^6 \end{vmatrix}\)is zero.

Part - D

Generic 5 - Answer All Question

Solve by using matrix inversion method:
2x + 5y = 1
3x + 2y = 7

Show that the matrices A=\(\left[ \begin{matrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{matrix} \right] \)and B=\(\left[ \begin{matrix} \frac { 4 }{ 5 } & -\frac { 2 }{ 5 } & -\frac { 1 }{ 5 } \\ -\frac { 1 }{ 5 } & \frac { 3 }{ 5 } & -\frac { 1 }{ 5 } \\ -\frac { 1 }{ 5 } & -\frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{matrix} \right] \) are inverses of each other.

An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investment is Rs. 70 more than the income from the third, find the amount of each investment by matrix method.
Show that the matrices \(A=\left[ \begin{matrix} 1 & 3 & 7 \\ 4 & 2 & 3 \\ 1 & 2 & 1 \end{matrix} \right] \)and \(B=\left[ \begin{matrix} \frac { -4 }{ 35 } & \frac { 11 }{ 35 } & \frac { -5 }{ 35 } \\ \frac { -1 }{ 35 } & \frac { -6 }{ 35 } & \frac { 25 }{ 35 } \\ \frac { 6 }{ 35 } & \frac { 1 }{ 35 } & \frac { -10 }{ 35 } \end{matrix} \right] \)are inverses of each other.

An economy produces only coal and steel. These two commodities serve as intermediate inputs in each other’s production. 0.4 tonne of steel and 0.7 tonne of coal are needed to produce a tonne of steel. Similarly 0.1 tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. Do you think that the system is viable? 2 and 5 labour days are required to produce a tonne s of coal and steel respectively. If economy needs 100 tonnes of coal and 50 tonnes of steel, calculate the gross output of the two commodities and the total labour days required.

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11th Business Maths Weekly Test -1 ( Matrices and Determinants )

MABS Institution

11th Business Maths Weekly Test -1 ( Matrices and Determinants )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

If A and B are non-singular matrices then, which of the following is incorrect?

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

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

If A \(=\begin{pmatrix} -1 & 2 \\ 1 & -4 \end{pmatrix}\) then A (adj A) is

The value of \(\begin{vmatrix} x & x^2 & -yz & 1 \\ y & y^2 & -zx & 1 \\ z & z^2 & -xy &1 \end{vmatrix}\) is

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

The inventor of input-output analysis is

Which of the following matrix has no inverse

If A is 3 x 3 matrix and |A|= 4, then |A-1| is equal to

If A = \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)such that ad - bc \(\neq\) 0 then A-1 is

Part - B

Generic 2 - Answer All Question

Show that \(\left[ \begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix} \right] \)is a singular matrix.

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

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

If \(A=\left| \begin{matrix} -2 & 6 \\ 3 & -9 \end{matrix} \right| \)then, find A-1

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

Solve: 2x+ 5y = 1 and 3x + 2y = 7 using matrix method.

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

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).

Using the properties of determinants, show that \(\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| \)=0

Without actual expansion show that the value of the determinant \(\begin{vmatrix}5 &5^2 &5^3 \\5^2 & 5^3 & 5^4\\5^4&5^5&5^6 \end{vmatrix}\)is zero.

Part - D

Generic 5 - Answer All Question

Solve by using matrix inversion method: 2x + 5y = 1 3x + 2y = 7

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

If A is 3 x 3 matrix and |A|= 4, then |A^-1| is equal to

If A = \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)such that ad - bc \(\neq\) 0 then A^-1 is

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

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

If \(A=\left| \begin{matrix} -2 & 6 \\ 3 & -9 \end{matrix} \right| \)then, find A^-1

Solve by using matrix inversion method:
2x + 5y = 1
3x + 2y = 7