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

MABS Institution

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

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

Suppose the inter-industry flow of the product of two sectors X and Yare given as under.

Production Sector Consumption Sector Domestic demand Gross output

X Y

X 15 10 10 35

Y 20 30 15 65

Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

Production Sector	Consumption Sector	Domestic demand	Gross output
	X	Y
X	15	10	10	35
Y	20	30	15	65

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

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

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

Solve\(\left| \begin{matrix} x-1 & x & x-2 \\ 0 & x-2 & x-3 \\ 0 & 0 & x-3 \end{matrix} \right| =0\)

Show that \(\left[ \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right] \)is non – singular.

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

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

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

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.

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

Evaluate \(\begin{vmatrix} 2 &-1 &-2 \\0 & 2 & -1\\3 & -5& 0 \end{vmatrix}.\)

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

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.

Part - B

Generic 3 - Answer All Question

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.

Prove that \(\begin{vmatrix} -a^{ 2 } & ab & ac \\ ab & -b^{ 2 } & bc \\ ac & bc & -c^{ 2 } \end{vmatrix}=4a^{ 2 }b^{ 2 }{ c }^{ 2 }\)

Write the minors and co-factors of the elements of \(\begin{vmatrix}5 & 3 \\-6 & 2\end{vmatrix}\)

if A=\(\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right] \) is such that A^T = A^-1, find \(\alpha\)

If\(A=\begin{bmatrix}1&3&3\\1&4&3\\1&3&4 \end{bmatrix}\)then verify that A (adj A) = |A| I and also find A^-1.

Show that \(\begin{vmatrix} a & a+b&a+b+c \\2a &3a+2b &4a+3b+2c\\3a&6a+3b&10a+6b+3c \end{vmatrix}=a^3.\)

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

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

Using co-factors of elements of second column evaluate \(\left| \begin{matrix} 6 & -1 & 5 \\ 3 & 0 & 4 \\ -2 & 7 & -3 \end{matrix} \right| \)

If A=\(\begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}\) and B=\(\begin{bmatrix} 4 & 6 \\ 3 & 2 \end{bmatrix}\), verify that (AB)^-1=B^-1A^-1

If X \(=\begin{bmatrix} 8 &-1&-3 \\-5 &1&2\\10&-1&-4 \end{bmatrix}\) and Y = \(\begin{bmatrix} 2 & 1 & -1\\0 & 2 & 1\\ 5& p & q \end{bmatrix}\) then, find p, q if Y = X^-1

If \({ A }^{ -1 }=\left[ \begin{matrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{matrix} \right] \) then, find A.

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

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

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

MABS Institution

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

Part - A

Generic 2 - Answer All Question

Suppose the inter-industry flow of the product of two sectors X and Yare given as under. Production Sector Consumption Sector Domestic demand Gross output X Y X 15 10 10 35 Y 20 30 15 65 Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

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

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

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

Solve\(\left| \begin{matrix} x-1 & x & x-2 \\ 0 & x-2 & x-3 \\ 0 & 0 & x-3 \end{matrix} \right| =0\)

Show that \(\left[ \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right] \)is non – singular.

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

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

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

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.

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

Evaluate \(\begin{vmatrix} 2 &-1 &-2 \\0 & 2 & -1\\3 & -5& 0 \end{vmatrix}.\)

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

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.

Part - B

Generic 3 - Answer All Question

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.

Prove that \(\begin{vmatrix} -a^{ 2 } & ab & ac \\ ab & -b^{ 2 } & bc \\ ac & bc & -c^{ 2 } \end{vmatrix}=4a^{ 2 }b^{ 2 }{ c }^{ 2 }\)

Write the minors and co-factors of the elements of \(\begin{vmatrix}5 & 3 \\-6 & 2\end{vmatrix}\)

if A=\(\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right] \) is such that AT = A-1, find \(\alpha\)

If\(A=\begin{bmatrix}1&3&3\\1&4&3\\1&3&4 \end{bmatrix}\)then verify that A (adj A) = |A| I and also find A-1.

Show that \(\begin{vmatrix} a & a+b&a+b+c \\2a &3a+2b &4a+3b+2c\\3a&6a+3b&10a+6b+3c \end{vmatrix}=a^3.\)

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

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

Using co-factors of elements of second column evaluate \(\left| \begin{matrix} 6 & -1 & 5 \\ 3 & 0 & 4 \\ -2 & 7 & -3 \end{matrix} \right| \)

If A=\(\begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}\) and B=\(\begin{bmatrix} 4 & 6 \\ 3 & 2 \end{bmatrix}\), verify that (AB)-1=B-1A-1

If X \(=\begin{bmatrix} 8 &-1&-3 \\-5 &1&2\\10&-1&-4 \end{bmatrix}\) and Y = \(\begin{bmatrix} 2 & 1 & -1\\0 & 2 & 1\\ 5& p & q \end{bmatrix}\) then, find p, q if Y = X-1

If \({ A }^{ -1 }=\left[ \begin{matrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{matrix} \right] \) then, find A.

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

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

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Tamilnadu Stateboardszs - Class

Suppose the inter-industry flow of the product of two sectors X and Yare given as under.

Production Sector Consumption Sector Domestic demand Gross output

X Y

X 15 10 10 35

Y 20 30 15 65

Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

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=\(\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right] \) is such that A^T = A^-1, find \(\alpha\)

If\(A=\begin{bmatrix}1&3&3\\1&4&3\\1&3&4 \end{bmatrix}\)then verify that A (adj A) = |A| I and also find A^-1.

If A=\(\begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}\) and B=\(\begin{bmatrix} 4 & 6 \\ 3 & 2 \end{bmatrix}\), verify that (AB)^-1=B^-1A^-1

If X \(=\begin{bmatrix} 8 &-1&-3 \\-5 &1&2\\10&-1&-4 \end{bmatrix}\) and Y = \(\begin{bmatrix} 2 & 1 & -1\\0 & 2 & 1\\ 5& p & q \end{bmatrix}\) then, find p, q if Y = X^-1