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Questions:

A total amount of Rs. 7,000 is deposited in three different savings bank accounts with annual interest rates of 5%, 8% and 81/2% respectively. The total annual interest from these three accounts is Rs. 550. Equal amounts have been deposited in 5% and 8% savings accounts. Find the amount deposited in each of the three accounts, with the help of matrices. Write the value.

Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to its 3, 2 and 1 students, respectively with a total award money of Rs. 2,200. School Q wants to spend Rs. 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as school P). If the total amount of award for one prize on each value is Rs. 1,200. Using matrices, find the award money for each value. Apart from the above these three values, suggest one more value which should be considered for award.

Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs. 1,000. School Q wants to spend Rs. 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs. 600. Using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Two schools A and B want to award their selected students on the values of Sincerity, Truthfulness and helpfulness. The school A wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs. 1,600. School B wants to spend Rs. 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs. 900. Using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

Two schools A and B want to award their selected teachers on the values of Honesty, Hard work and Regularity. School A wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to 3, 2 and 1 teachers with a total award money of Rs.1.28 lakh. School B wants to spend Rs. 1·54 lakh to award its 4, 1 and 3 teachers on the respective values (by giving the same award money for the three values as before). If the total amount of award for one prize on each value is Rs. 57,000, using matrices, find the award money for each value.

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Answers Key:

Let Rs. X, Y and z be invested in saving accounts at the rate 5%, 8% and 81/2% respectively. Then the system of equations is
x + y + z = 7000
\(\frac { 5x }{ 100 } +\frac { 8y }{ 100 } +\frac { 17z }{ 100 } =550\)
\(\Rightarrow\) 10x + 16y + 17z = 1,10,00
x - y = 0
Matrix equation is
\(\left[ \begin{matrix} 1 & 1 & 1 \\ 10 & 16 & 17 \\ 1 & -1 & 0 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 7,000 \\ 1,10,000 \\ 0 \end{matrix} \right] \)
i.e., AX = B
|A| = 1(0+17)-1(0-17)+1(-10-16)
= 8 \(\ne\)0
A^-1 exists.
Cofactors are:
\(\left[ \begin{matrix} { A }_{ 11 }=17, & { A }_{ 12 }=17, & { A }_{ 13 }=-26 \\ { A }_{ 21 }=-1, & { A }_{ 22 }=-1, & { A }_{ 23 }=2 \\ { A }_{ 31 }=1, & { A }_{ 32 }=-7 & { A }_{ 33 }=6 \end{matrix} \right] \)
X=A^-1B
\(\Rightarrow \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\frac { 1 }{ 8 } \left[ \begin{matrix} 17 & -1 & 1 \\ 17 & -1 & -7 \\ -26 & 2 & 6 \end{matrix} \right] \left[ \begin{matrix} 7,000 \\ 1,10,000 \\ 0 \end{matrix} \right] \)
\(\Rightarrow \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\frac { 1 }{ 8 } \left[ \begin{matrix} 1,125 \\ 1,125 \\ 4,750 \end{matrix} \right] \)
\(\Rightarrow\) x = 1,125, y = 1,125, z = 4,750
\(\therefore\)Amount deposited in each type of account is Rs. 1125, Rs. 1125 and Rs. 4750 respectively.
A penny saved is a penny earned.

Here, 3x + 2y + Z = 2,200
4x + y + 3z = 3,100
x + y + z = 1,200
Matrix equation is
\(\left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 2,200 \\ 3,100 \\ 1,200 \end{matrix} \right] \quad or\quad AX=B\)
|A| = 3(-2)-2(1)+1(3)
= -5 \(\ne\)0
\(\therefore\) X = A^-1B
cofactors are:
\(\left[ \begin{matrix} { A }_{ 11 }=-2 & { A }_{ 12 }=-1 & { A }_{ 13 }=3 \\ { A }_{ 21 }=-1 & { A }_{ 22 }=2 & { A }_{ 23 }=-1 \\ { A }_{ 31 }=5 & { A }_{ 32 }=-5 & { A }_{ 33 }=-5 \end{matrix} \right] \)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =-\frac { 1 }{ 5 } \left[ \begin{matrix} -2 & -1 & 5 \\ -1 & 2 & -5 \\ 3 & -1 & -5 \end{matrix} \right] \left[ \begin{matrix} 2,200 \\ 3,100 \\ 1,200 \end{matrix} \right] \)
x = 300, y = 400, z = 500
One more value punctuality, honesty etc.

3x + 2y + z = 1000
4x + y + 3z = 1500
x + y + z = 600
\(\left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 1000 \\ 1500 \\ 600 \end{matrix} \right] \ or\ AX=B\)
|A| = 3(-2)-2(1)+1(3)
= -5 \(\ne\)0
\(\therefore X={ A }^{ -1 }B\)
Cofactors are
\({ A }_{ 11 }=-2, { A }_{ 12 }=-1, { A }_{ 13 }=3 \)
\({ A }_{ 21 }=-1, { A }_{ 22 }=2, { A }_{ 23 }=-1 \)
\({ A }_{ 31 }=5, { A }_{ 32 }=-5, { A }_{ 33 }=-5 \)
\(\therefore \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =-\frac { 1 }{ 5 } \left[ \begin{matrix} -2 & -1 & 5 \\ -1 & 2 & -5 \\ 3 & -1 & -5 \end{matrix} \right] \left[ \begin{matrix} 1000 \\ 1500 \\ 600 \end{matrix} \right] \)
x = 100, y = 200 and z = 300
i.e., Rs. 100 for discipline, Rs. 200 for politeness and Rs. 300 for punctuality.
Value: One more value like sincerity, truthfulness etc.

3x + 2y + z = 1,600
4x + Y + 3z = 2,300
x + y + z = 900
\(\therefore \left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 1,600 \\ 2,300 \\ 900 \end{matrix} \right] \ or\ AX=B\)
|A| = 3(-2)-2(1)+1(3)
= -5 \(\ne\)0
Cofactors are:
\({ A }_{ 11 }=-2, { A }_{ 12 }=-1, { A }_{ 13 }=3 \)
\({ A }_{ 21 }=-1, { A }_{ 22 }=2, { A }_{ 23 }=-1 \)
\({ A }_{ 31 }=5, { A }_{ 32 }=-5, { A }_{ 33 }=-5 \)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =-\frac { 1 }{ 5 } \left[ \begin{matrix} -2 & -1 & 5 \\ -1 & 2 & -5 \\ 3 & -1 & -5 \end{matrix} \right] \left[ \begin{matrix} 1,600 \\ 2,300 \\ 900 \end{matrix} \right] \)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 200 \\ 300 \\ 400 \end{matrix} \right] \)
x = 200, y = 300 and z = 400
i.e., Rs. 200 for sincerity, Rs. 300 for truthfullness and Rs. 400 for helpfulness.
Value: One more value like honesty, kindness etc.

Let x, y and z be the values of honesty, hard work and regularity respectively. Then the system of equations is
3x + 2y + z = 1.28
4x + y + 3z= 1.54
x + y + z = 0.57
Matrix equation is
\(\left[ \begin{matrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 1.28 \\ 1.54 \\ 0.57 \end{matrix} \right] \)
i.e., AX=B
\(|A|=\left| \begin{matrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{matrix} \right| \)
= 3(1-3)-2(4-3)+1(4-1)
= -5 \(\ne\)0
A^-1 exists.
adj A=\({ \left[ \begin{matrix} -2 & -1 & 3 \\ -1 & 2 & -1 \\ 5 & -5 & -5 \end{matrix} \right] }^{ T }\)
\(=\left[ \begin{matrix} -2 & -1 & 5 \\ -1 & 2 & -5 \\ 3 & -1 & -5 \end{matrix} \right] \)
\(X={ A }^{ -1 }B\)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =-\frac { 1 }{ 5 } \left[ \begin{matrix} -2 & -1 & 5 \\ -1 & 2 & -5 \\ 3 & -1 & -5 \end{matrix} \right] \left[ \begin{matrix} 1.28 \\ 1.54 \\ 0.57 \end{matrix} \right] \)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 0.25 \\ 0.21 \\ 0.11 \end{matrix} \right] \)
\(\therefore\) x = 25,000, y = 21,000, z = 11,000
i.e., Rs. 25,000 for honesty, Rs. 21,000 for hard work and Rs. 11,000 for regularity.
Value: One more value like sincerity kindness etc.