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11th Business Maths Weekly Test - 3 ( Matrices and Determinants )

MABS Institution

11th Business Maths Weekly Test - 3 ( Matrices and Determinants )-Aug 2020

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

If A = \(\begin{vmatrix}cos \theta & sin \theta \\ -sin \theta&cons\theta \end{vmatrix}\) then |2A| is equal to

4 cos 2 \(\theta\)
4
2
1

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

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

The number of Hawkins-Simon conditions for the viability of an input - output analysis is

1
3
4
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

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 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

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

0
2
1
3

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

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

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

If \(\triangle=\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}\) and A_ij is cofactor of a_ij, then value of \(\triangle\) is given by

a₁₁ A₃₁ + a₁₂ A₃₂ + a₁₃ A₃₃
a₁₁ A₁₁ + a₁₂A₂₁ + a₁₃ A₃₁
a₂₁ A₁₁ + a₂₂ A₁₂ + a₂₃ A₁₃
a₁₁ A₁₁ + a₂₁ A₂₁ + a₃₁ A₃₁

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

81
27
3
9

If \(\begin{vmatrix} 4 & 3 \\ 3 & 1 \end{vmatrix}=-5\) then value of \(\begin{vmatrix} 20 & 15 \\ 15 & 5 \end{vmatrix}\) is

-5
-125
-25
0

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

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

The inventor of input-output analysis is

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

Part - B

Generic 2 - Answer All Question

Find the minors and cofactors of all the elements of the following determinants
\(\begin{vmatrix}5&20\\ 0&-1 \end{vmatrix}\)

Prove that \(\begin{vmatrix} {1\over a}&bc&b+c\\{1\over b}&ca&c+a\\{1\over c}&ab&a+b \end{vmatrix}=0\)

Solve:\(\begin{vmatrix}7&4&11\\-3&5&x\\-x&3&1 \end{vmatrix}=0\)

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

The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.50 & 0.30 \\ 0.41 & 0.33 \end{bmatrix}\). Test whether the system is viable as per Hawkins Simon conditions.

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

Find |AB| if \(A=\begin{bmatrix} 3&-1\\2&1 \end{bmatrix}and \begin{bmatrix} 3&0\\1&-2 \end{bmatrix}\)

Find adj A for \(A=\left[ \begin{matrix} 2 & 3 \\ 1 & 4 \end{matrix} \right] \)

Show that\(\left| \overset { x }{ 2x\underset { a }{ + } 2a } \quad \overset { y }{ 2y\underset { b }{ + } 2b } \quad \overset { z }{ 2z\underset { c }{ + } 2c } \right| =0\)

Solve: \(\begin{vmatrix}2& x&3\\4&1&6\\1&2&7 \end{vmatrix}=0\)

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