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.

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 adaptable new techniques, careful and alert in difficult situations and keeping calm in tense situations respectively.
Then the system of equations is
2x + 4y + 3z = 29,000 
5x + 2y + 3z = 30,500 
x + y + z = 9,500
Matrix equation is
\(\left[ \begin{matrix} 2 & 4 & 3 \\ 5 & 2 & 3 \\ 1 & 1 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 29000 \\ 30500 \\ 9500 \end{matrix} \right] \)
i.e, AX = B
\(|A|=\left| \begin{matrix} 2 & 4 & 3 \\ 5 & 2 & 3 \\ 1 & 1 & 1 \end{matrix} \right| \)
= 2(2-3)-495-3)+3(5-2)
= -1 \(\ne\) 0
\(\therefore\) A^-1 exists.
adj A = \({ \left[ \begin{matrix} -1 & -2 & 3 \\ -2 & -1 & 2 \\ 3 & 2 & -16 \end{matrix} \right] }^{ T }\)
\(=\left[ \begin{matrix} -1 & -1 & 6 \\ -2 & -1 & 9 \\ 3 & 2 & -16 \end{matrix} \right] \)
\(X={ A }^{ -1 }B\)
\(=-1\left[ \begin{matrix} -29,000-30,500+57,000 \\ -58,000-30,500+85,500 \\ 87,00+61,000-15,2,000 \end{matrix} \right] \)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 2,500 \\ 3,000 \\ 4,000 \end{matrix} \right] \)
x = 2,500, y = 3,000, z = 4,000
i.e., Rs. 2,500 for adaptable n.ew technique Rs. 3,000 for careful and alert in difficult situations and Rs. 4,000 for keeping calm in tense situations,
Value: Award should'be given to a person who truly deserves it.

Let the award money for the values of honesty, regularity and hard work be Rs. x Rs. y and Rs. z respectively. Then the. system of equations is
x + y + z = 6,000 
x + 3z = 11,000
x - 2y + z = 0
Matrix equation is
\(\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 6,000 \\ 11,000 \\ 0 \end{matrix} \right] \)
i.e., AX = B
\(|A|=\quad \left| \begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{matrix} \right| \)
= 1(0+6)-1(1-3)+1(-2-0)
= 6 \(\ne\)0
\(\therefore\) A^-1 exists.
adj A = \({ \left[ \begin{matrix} 6 & 2 & -2 \\ -3 & 0 & 3 \\ 3 & -2 & -1 \end{matrix} \right] }^{ T }\)
\(=\left[ \begin{matrix} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & -1 \end{matrix} \right] \)
\(X={ A }^{ -1 }B\)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\frac { 1 }{ 6 } \left[ \begin{matrix} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & -1 \end{matrix} \right] \left[ \begin{matrix} 6,000 \\ 11,000 \\ 0 \end{matrix} \right] \quad \)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 500 \\ 2,000 \\ 3,500 \end{matrix} \right] \)
x = 500, y = 2,000, z = 3,500 
Hence, award money given for the value of Honesty = Rs. 500, award money given for the value of Regularity = Rs. 2,000 and, award money given. for the value of Hard-work = Rs. 3,500.
Value: The school must include the value of Obedience for the awards.

\({ A }^{ -1 }=\left[ \begin{matrix} 1/a & 0 & 0 \\ 0 & 1/b & 0 \\ 0 & 0 & 1/c \end{matrix} \right] \)
The inverse is impossible. It is possible when a, b, c are all real and\(\ne\)0, otherwise |A| becomes singular
i.e., |A| = 0
and \({ A }^{ -1 }=\frac { adj\quad A }{ |A| } \)
will become meaningless.
Value: Anybody can progress till he continues working, otherwise his progress will be halted.