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First Year Math Matrices and Determinants 1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(vii)\[\begin{array}{l}2 x-2 y=4 \\-5 x-2 y=-10\end{array}\]


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Class 9Class 10First YearSecond Year
1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(vii)\[\begin{array}{l}2 x-2 y=4 \\-5 x-2 y=-10\end{array}\]

Q.20 If A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] then adj A equals :(a) \left[\begin{array}{rr}d & b \\ -c & a\end{array}\right] (b) \left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right] (c) \left[\begin{array}{ll}d & b \\ c & a\end{array}\right] (d) \left[\begin{array}{rr}d & -b \\ c & a\end{array}\right] Gujranwala Board 2007
Q.20 If  A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]  then adj  A  equals :(a)  \left[\begin{array}{rr}d & b \\ -c & a\end{array}\right] (b)  \left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right] (c)  \left[\begin{array}{ll}d & b \\ c & a\end{array}\right] (d)  \left[\begin{array}{rr}d & -b \\ c & a\end{array}\right] Gujranwala Board 2007

Q.20 If A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] then adj A equals :(a) \left[\begin{array}{rr}d & b \\ -c & a\end{array}\right] (b) \left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right] (c) \left[\begin{array}{ll}d & b \\ c & a\end{array}\right] (d) \left[\begin{array}{rr}d & -b \\ c & a\end{array}\right] Gujranwala Board 2007

Evaluate the following determinants using their properties.8. \left|\begin{array}{lll}x & y & z \\ z & x & y \\ y & z & x\end{array}\right|
Evaluate the following determinants using their properties.8.  \left|\begin{array}{lll}x & y & z \\ z & x & y \\ y & z & x\end{array}\right|

Evaluate the following determinants using their properties.8. \left|\begin{array}{lll}x & y & z \\ z & x & y \\ y & z & x\end{array}\right|

Evaluate the following determinants:6. \left|\begin{array}{lll}1 & 1 & \omega \\ 1 & 1 & \omega^{2} \\ \omega & \omega^{2} & 1\end{array}\right| where \omega is a complex cube root of unity.
Evaluate the following determinants:6.  \left|\begin{array}{lll}1 & 1 & \omega \\ 1 & 1 & \omega^{2} \\ \omega & \omega^{2} & 1\end{array}\right| where  \omega  is a complex cube root of unity.

Evaluate the following determinants:6. \left|\begin{array}{lll}1 & 1 & \omega \\ 1 & 1 & \omega^{2} \\ \omega & \omega^{2} & 1\end{array}\right| where \omega is a complex cube root of unity.

17. Let \mathrm{A}=\left[\begin{array}{ccc}1 & 3 & 0 \\ -1 & 2 & 1 \\ 0 & 0 & 2\end{array}\right] \mathrm{B}=\left[\begin{array}{ccc}2 & 3 & 4 \\ 1 & 2 & 3 \\ -1 & 1 & 2\end{array}\right] and C=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 5 & 8 \\ 2 & 7 & 6\end{array}\right] Show that :(V) \mathrm{A}+\mathrm{B}=\mathrm{B}+\mathrm{A}^{-7}=\pm=
17. Let  \mathrm{A}=\left[\begin{array}{ccc}1 & 3 & 0 \\ -1 & 2 & 1 \\ 0 & 0 & 2\end{array}\right] \mathrm{B}=\left[\begin{array}{ccc}2 & 3 & 4 \\ 1 & 2 & 3 \\ -1 & 1 & 2\end{array}\right]  and  C=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 5 & 8 \\ 2 & 7 & 6\end{array}\right] Show that :(V)  \mathrm{A}+\mathrm{B}=\mathrm{B}+\mathrm{A}^{-7}=\pm=

17. Let \mathrm{A}=\left[\begin{array}{ccc}1 & 3 & 0 \\ -1 & 2 & 1 \\ 0 & 0 & 2\end{array}\right] \mathrm{B}=\left[\begin{array}{ccc}2 & 3 & 4 \\ 1 & 2 & 3 \\ -1 & 1 & 2\end{array}\right] and C=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 5 & 8 \\ 2 & 7 & 6\end{array}\right] Show that :(V) \mathrm{A}+\mathrm{B}=\mathrm{B}+\mathrm{A}^{-7}=\pm=

(vi) Product of \left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{c}2 \\ -1\end{array}\right] is(a) [2 x+y] (b) [x-2 y] (c) [2 x-y] (d) [x+2 y]
(vi) Product of  \left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{c}2 \\ -1\end{array}\right]  is(a)  [2 x+y] (b)  [x-2 y] (c)  [2 x-y] (d)  [x+2 y]

(vi) Product of \left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{c}2 \\ -1\end{array}\right] is(a) [2 x+y] (b) [x-2 y] (c) [2 x-y] (d) [x+2 y]

Q.14 \left[\begin{array}{lll}0 & 0 & 0\end{array}\right] is(a) scalar matrix(b) diagonal matrix(c) identity matrix(d) null matrix
Q.14  \left[\begin{array}{lll}0 & 0 & 0\end{array}\right]  is(a) scalar matrix(b) diagonal matrix(c) identity matrix(d) null matrix

Q.14 \left[\begin{array}{lll}0 & 0 & 0\end{array}\right] is(a) scalar matrix(b) diagonal matrix(c) identity matrix(d) null matrix

Evaluate the following determinants using their properties.10. \left|\begin{array}{ccc}x+y & x & y \\ x & y & x+y \\ y & x+y & x\end{array}\right|
Evaluate the following determinants using their properties.10.  \left|\begin{array}{ccc}x+y & x & y \\ x & y & x+y \\ y & x+y & x\end{array}\right|

Evaluate the following determinants using their properties.10. \left|\begin{array}{ccc}x+y & x & y \\ x & y & x+y \\ y & x+y & x\end{array}\right|

Q.32 If |A| is the determinant of a square matrix A then |A| is(a) always positive(b) modulus of A (c) always negative(d) may be nonnegative or negative
Q.32 If  |A|  is the determinant of a square matrix  A  then  |A|  is(a) always positive(b) modulus of  A (c) always negative(d) may be nonnegative or negative

Q.32 If |A| is the determinant of a square matrix A then |A| is(a) always positive(b) modulus of A (c) always negative(d) may be nonnegative or negative

4.\[\begin{array}{l}l f=\left[\begin{array}{ccc}4 & 2 & 0 \\5 & 6 & 7 \\-3 & 1 & 9\end{array}\right] B=\left[\begin{array}{lll}2 & 3 & 1 \\1 & 1 & 1\end{array}\right] \\C=\left[\begin{array}{cc}-2 & -3 \\-4 & 0 \\1 & 3\end{array}\right]\end{array}\]then wherever possible compute the following :(v) CB
4.\[\begin{array}{l}l f=\left[\begin{array}{ccc}4 & 2 & 0 \\5 & 6 & 7 \\-3 & 1 & 9\end{array}\right] B=\left[\begin{array}{lll}2 & 3 & 1 \\1 & 1 & 1\end{array}\right] \\C=\left[\begin{array}{cc}-2 & -3 \\-4 & 0 \\1 & 3\end{array}\right]\end{array}\]then wherever possible compute the following :(v) CB

4.\[\begin{array}{l}l f=\left[\begin{array}{ccc}4 & 2 & 0 \\5 & 6 & 7 \\-3 & 1 & 9\end{array}\right] B=\left[\begin{array}{lll}2 & 3 & 1 \\1 & 1 & 1\end{array}\right] \\C=\left[\begin{array}{cc}-2 & -3 \\-4 & 0 \\1 & 3\end{array}\right]\end{array}\]then wherever possible compute the following :(v) CB

1. Find the inverse wherever possible of each of the following matrices by the adjoint method.(ix) \left[\begin{array}{lll}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right] where \omega is a complex cube root of unity.
1. Find the inverse wherever possible of each of the following matrices by the adjoint method.(ix)  \left[\begin{array}{lll}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right] where  \omega  is a complex cube root of unity.

1. Find the inverse wherever possible of each of the following matrices by the adjoint method.(ix) \left[\begin{array}{lll}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right] where \omega is a complex cube root of unity.

Q.19 The transpose of a row matrix is a(a) column matrix(b) diagonal matrix
Q.19 The transpose of a row matrix is a(a) column matrix(b) diagonal matrix

Q.19 The transpose of a row matrix is a(a) column matrix(b) diagonal matrix

Evaluate the following determinants using their properties.9. \left|\begin{array}{lll}1 & x^{3} & x \\ 1 & y^{3} & y \\ 1 & z^{2} & z\end{array}\right|
Evaluate the following determinants using their properties.9.  \left|\begin{array}{lll}1 & x^{3} & x \\ 1 & y^{3} & y \\ 1 & z^{2} & z\end{array}\right|

Evaluate the following determinants using their properties.9. \left|\begin{array}{lll}1 & x^{3} & x \\ 1 & y^{3} & y \\ 1 & z^{2} & z\end{array}\right|

Q.62 If A=\left[\begin{array}{c}2 \\ -1\end{array}\right] and B=\left[\begin{array}{ll}5 & 0\end{array}\right] then A B is equal to :(a) \left[\begin{array}{rr}10 & 0 \\ -5 & 0\end{array}\right] (b) \left[\begin{array}{c}10 \\ 0\end{array}\right] (c) \left[\begin{array}{ll}10 & 0\end{array}\right] (d) \left[\begin{array}{c}10 \\ -5\end{array}\right] Falsalabad Board 2012
Q.62 If  A=\left[\begin{array}{c}2 \\ -1\end{array}\right]  and  B=\left[\begin{array}{ll}5 & 0\end{array}\right]  then  A B  is equal to :(a)  \left[\begin{array}{rr}10 & 0 \\ -5 & 0\end{array}\right] (b)  \left[\begin{array}{c}10 \\ 0\end{array}\right] (c)  \left[\begin{array}{ll}10 & 0\end{array}\right] (d)  \left[\begin{array}{c}10 \\ -5\end{array}\right] Falsalabad Board 2012

Q.62 If A=\left[\begin{array}{c}2 \\ -1\end{array}\right] and B=\left[\begin{array}{ll}5 & 0\end{array}\right] then A B is equal to :(a) \left[\begin{array}{rr}10 & 0 \\ -5 & 0\end{array}\right] (b) \left[\begin{array}{c}10 \\ 0\end{array}\right] (c) \left[\begin{array}{ll}10 & 0\end{array}\right] (d) \left[\begin{array}{c}10 \\ -5\end{array}\right] Falsalabad Board 2012

Evaluate the following determinants:1. \left|\begin{array}{rrr}1 & 1 & 1 \\ -1 & -1 & 0 \\ -1 & 0 & 1\end{array}\right|
Evaluate the following determinants:1.  \left|\begin{array}{rrr}1 & 1 & 1 \\ -1 & -1 & 0 \\ -1 & 0 & 1\end{array}\right|

Evaluate the following determinants:1. \left|\begin{array}{rrr}1 & 1 & 1 \\ -1 & -1 & 0 \\ -1 & 0 & 1\end{array}\right|

9. Let A=\left[\begin{array}{lll}2 & 1 & 1 \\ 3 & 1 & 0 \\ 0 & 1 & 2\end{array}\right] and B=\left[\begin{array}{ll}0 & 1 \\ 1 & 2 \\ 2 & 0\end{array}\right] . Find:(iv) \mathrm{B}^{8} \cdot \mathrm{A}^{2}
9. Let  A=\left[\begin{array}{lll}2 & 1 & 1 \\ 3 & 1 & 0 \\ 0 & 1 & 2\end{array}\right]  and  B=\left[\begin{array}{ll}0 & 1 \\ 1 & 2 \\ 2 & 0\end{array}\right] . Find:(iv)  \mathrm{B}^{8} \cdot \mathrm{A}^{2}

9. Let A=\left[\begin{array}{lll}2 & 1 & 1 \\ 3 & 1 & 0 \\ 0 & 1 & 2\end{array}\right] and B=\left[\begin{array}{ll}0 & 1 \\ 1 & 2 \\ 2 & 0\end{array}\right] . Find:(iv) \mathrm{B}^{8} \cdot \mathrm{A}^{2}

1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(vii)\[\begin{array}{l}2 x-2 y=4 \\-5 x-2 y=-10\end{array}\]
1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(vii)\[\begin{array}{l}2 x-2 y=4 \\-5 x-2 y=-10\end{array}\]
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1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(vii)\[\begin{array}{l}2 x-2 y=4 \\-5 x-2 y=-10\end{array}\]

Solve the following word problems by using(i) matrix inversion method (ii) Crammers rule.6 Two cars that are 600 \mathrm{~km} apart are moving towards each other. Their speeds differ by 6 \mathrm{~km} per hour and the cars are 123 \mathrm{~km} apart after 4 \frac{1}{2} hours. Find the speed of each car.
Solve the following word problems by using(i) matrix inversion method (ii) Crammers rule.6 Two cars that are  600 \mathrm{~km}  apart are moving towards each other. Their speeds differ by  6 \mathrm{~km}  per hour and the cars are  123 \mathrm{~km}  apart after  4 \frac{1}{2}  hours. Find the speed of each car.

Solve the following word problems by using(i) matrix inversion method (ii) Crammers rule.6 Two cars that are 600 \mathrm{~km} apart are moving towards each other. Their speeds differ by 6 \mathrm{~km} per hour and the cars are 123 \mathrm{~km} apart after 4 \frac{1}{2} hours. Find the speed of each car.

8. Let A=\left[\begin{array}{ccc}-2 & 1 & 0 \\ -1 & 4 & 3 \\ 0 & 8 & 5\end{array}\right] and X=\left[\begin{array}{ccc}2 & 1 & -1 \\ -3 & 2 & -4 \\ 5 & 4 & 0\end{array}\right] Find:(ii) \lambda I-A where \lambda is any scalar and I is a unit matrix of order 3 .
8. Let  A=\left[\begin{array}{ccc}-2 & 1 & 0 \\ -1 & 4 & 3 \\ 0 & 8 & 5\end{array}\right]  and  X=\left[\begin{array}{ccc}2 & 1 & -1 \\ -3 & 2 & -4 \\ 5 & 4 & 0\end{array}\right] Find:(ii)   \lambda I-A where  \lambda  is any scalar and I is a unit matrix of order  3 .

8. Let A=\left[\begin{array}{ccc}-2 & 1 & 0 \\ -1 & 4 & 3 \\ 0 & 8 & 5\end{array}\right] and X=\left[\begin{array}{ccc}2 & 1 & -1 \\ -3 & 2 & -4 \\ 5 & 4 & 0\end{array}\right] Find:(ii) \lambda I-A where \lambda is any scalar and I is a unit matrix of order 3 .

Q.58 If \left|\begin{array}{cc}-1 & 3 \\ x & 1\end{array}\right|=0 then x= (a) 3(b) -3 (c) \frac{1}{3} (d) -\frac{1}{3} Lahore Board 2015; Multan Board 2004
Q.58 If  \left|\begin{array}{cc}-1 & 3 \\ x & 1\end{array}\right|=0  then  x= (a) 3(b)  -3 (c)  \frac{1}{3} (d)  -\frac{1}{3} Lahore Board 2015; Multan Board 2004
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Q.58 If \left|\begin{array}{cc}-1 & 3 \\ x & 1\end{array}\right|=0 then x= (a) 3(b) -3 (c) \frac{1}{3} (d) -\frac{1}{3} Lahore Board 2015; Multan Board 2004

1. Specily the type of each of the following matrices.(1) \left[\begin{array}{ccc}3 & 0 & 0 \\ 0 & -\frac{1}{2} & 0 \\ 0 & 0 & \sqrt{5}\end{array}\right]
1. Specily the type of each of the following matrices.(1)  \left[\begin{array}{ccc}3 & 0 & 0 \\ 0 & -\frac{1}{2} & 0 \\ 0 & 0 & \sqrt{5}\end{array}\right]
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1. Specily the type of each of the following matrices.(1) \left[\begin{array}{ccc}3 & 0 & 0 \\ 0 & -\frac{1}{2} & 0 \\ 0 & 0 & \sqrt{5}\end{array}\right]

Show that :11. \left|\begin{array}{lll}x+2 y & x+6 y & x+4 y \\ x+3 y & x+7 y & x+5 y \\ x+4 y & x+8 y & x+6 y\end{array}\right|=0
Show that :11.  \left|\begin{array}{lll}x+2 y & x+6 y & x+4 y \\ x+3 y & x+7 y & x+5 y \\ x+4 y & x+8 y & x+6 y\end{array}\right|=0
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Show that :11. \left|\begin{array}{lll}x+2 y & x+6 y & x+4 y \\ x+3 y & x+7 y & x+5 y \\ x+4 y & x+8 y & x+6 y\end{array}\right|=0

1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(i) 2 x-2 y=4 3 x+2 y=6
1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(i)  2 x-2 y=4  3 x+2 y=6
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1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(i) 2 x-2 y=4 3 x+2 y=6

Q.36 If A=\left[\begin{array}{ll}1 & 7 \\ 6 & 4\end{array}\right] then |A| is equal to :(a) 46(b) -46 (c) 38(d) -38
Q.36 If  A=\left[\begin{array}{ll}1 & 7 \\ 6 & 4\end{array}\right]  then  |A|  is equal to :(a) 46(b)  -46 (c) 38(d)  -38
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Q.36 If A=\left[\begin{array}{ll}1 & 7 \\ 6 & 4\end{array}\right] then |A| is equal to :(a) 46(b) -46 (c) 38(d) -38

Example.Let A=\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 6 & 8 \\ -1 & 2 & -3\end{array}\right] and B=\left[\begin{array}{ccc}2 & 3 & 4 \\ -4 & 6 & -8 \\ 1 & 2 & 5\end{array}\right] Then(iii)\[\begin{aligned}A B &=\left[\begin{array}{ccc}2-8+3 & 3+12+6 & 4-16+15 \\8-24+8 & 12+36+16 & 16-48+40 \\-2-8-3 & -3+12-6 & -4-16-15\end{array}\right] \\&=\left[\begin{array}{ccc}-3 & 21 & 3 \\-8 & 64 & 8 \\-13 & 3 & -35\end{array}\right]\end{aligned}\]
Example.Let  A=\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 6 & 8 \\ -1 & 2 & -3\end{array}\right]  and  B=\left[\begin{array}{ccc}2 & 3 & 4 \\ -4 & 6 & -8 \\ 1 & 2 & 5\end{array}\right] Then(iii)\[\begin{aligned}A B &=\left[\begin{array}{ccc}2-8+3 & 3+12+6 & 4-16+15 \\8-24+8 & 12+36+16 & 16-48+40 \\-2-8-3 & -3+12-6 & -4-16-15\end{array}\right] \\&=\left[\begin{array}{ccc}-3 & 21 & 3 \\-8 & 64 & 8 \\-13 & 3 & -35\end{array}\right]\end{aligned}\]
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Example.Let A=\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 6 & 8 \\ -1 & 2 & -3\end{array}\right] and B=\left[\begin{array}{ccc}2 & 3 & 4 \\ -4 & 6 & -8 \\ 1 & 2 & 5\end{array}\right] Then(iii)\[\begin{aligned}A B &=\left[\begin{array}{ccc}2-8+3 & 3+12+6 & 4-16+15 \\8-24+8 & 12+36+16 & 16-48+40 \\-2-8-3 & -3+12-6 & -4-16-15\end{array}\right] \\&=\left[\begin{array}{ccc}-3 & 21 & 3 \\-8 & 64 & 8 \\-13 & 3 & -35\end{array}\right]\end{aligned}\]

Evaluate the following determinants using their properties.7. \left|\begin{array}{ccc}a+b+2 c & b & a \\ c & b & b+c+2 a \\ c & c+a+2 b & a\end{array}\right|
Evaluate the following determinants using their properties.7.  \left|\begin{array}{ccc}a+b+2 c & b & a \\ c & b & b+c+2 a \\ c & c+a+2 b & a\end{array}\right|
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Evaluate the following determinants using their properties.7. \left|\begin{array}{ccc}a+b+2 c & b & a \\ c & b & b+c+2 a \\ c & c+a+2 b & a\end{array}\right|

18. Prove the following identities :\[\left\{\left[\begin{array}{lll}1 & \omega & \omega^{2} \\\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega\end{array}\right]+\left[\begin{array}{lll}\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega \\\omega & \omega^{2} & 1\end{array}\right]\right\}\left[\begin{array}{l}1 \\\omega \\\omega^{2}\end{array}\right]=\left[\begin{array}{l}0 \\0 \\0\end{array}\right]\]where \omega is a complex cube root of unity.
18. Prove the following identities :\[\left\{\left[\begin{array}{lll}1 & \omega & \omega^{2} \\\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega\end{array}\right]+\left[\begin{array}{lll}\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega \\\omega & \omega^{2} & 1\end{array}\right]\right\}\left[\begin{array}{l}1 \\\omega \\\omega^{2}\end{array}\right]=\left[\begin{array}{l}0 \\0 \\0\end{array}\right]\]where  \omega  is a complex cube root of unity.
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18. Prove the following identities :\[\left\{\left[\begin{array}{lll}1 & \omega & \omega^{2} \\\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega\end{array}\right]+\left[\begin{array}{lll}\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega \\\omega & \omega^{2} & 1\end{array}\right]\right\}\left[\begin{array}{l}1 \\\omega \\\omega^{2}\end{array}\right]=\left[\begin{array}{l}0 \\0 \\0\end{array}\right]\]where \omega is a complex cube root of unity.

6. Verify that if \mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] \mathrm{B}=\left[\begin{array}{ll}1 & 1 \\ 2 & 0\end{array}\right] then(i) \left(A^{t}\right)^{t}=A
6. Verify that if  \mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] \mathrm{B}=\left[\begin{array}{ll}1 & 1 \\ 2 & 0\end{array}\right]  then(i)  \left(A^{t}\right)^{t}=A
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6. Verify that if \mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] \mathrm{B}=\left[\begin{array}{ll}1 & 1 \\ 2 & 0\end{array}\right] then(i) \left(A^{t}\right)^{t}=A

4.\[\begin{array}{l}l f=\left[\begin{array}{ccc}4 & 2 & 0 \\5 & 6 & 7 \\-3 & 1 & 9\end{array}\right] B=\left[\begin{array}{lll}2 & 3 & 1 \\1 & 1 & 1\end{array}\right] \\C=\left[\begin{array}{cc}-2 & -3 \\-4 & 0 \\1 & 3\end{array}\right]\end{array}\]then wherever possible compute the following :(im) BC
4.\[\begin{array}{l}l f=\left[\begin{array}{ccc}4 & 2 & 0 \\5 & 6 & 7 \\-3 & 1 & 9\end{array}\right] B=\left[\begin{array}{lll}2 & 3 & 1 \\1 & 1 & 1\end{array}\right] \\C=\left[\begin{array}{cc}-2 & -3 \\-4 & 0 \\1 & 3\end{array}\right]\end{array}\]then wherever possible compute the following :(im) BC
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4.\[\begin{array}{l}l f=\left[\begin{array}{ccc}4 & 2 & 0 \\5 & 6 & 7 \\-3 & 1 & 9\end{array}\right] B=\left[\begin{array}{lll}2 & 3 & 1 \\1 & 1 & 1\end{array}\right] \\C=\left[\begin{array}{cc}-2 & -3 \\-4 & 0 \\1 & 3\end{array}\right]\end{array}\]then wherever possible compute the following :(im) BC

3. Find the following producte.(iv) \left[\begin{array}{ll}6 & -0\end{array}\right]\left[\begin{array}{l}4 \\ 0\end{array}\right]
3. Find the following producte.(iv)  \left[\begin{array}{ll}6 & -0\end{array}\right]\left[\begin{array}{l}4 \\ 0\end{array}\right]
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3. Find the following producte.(iv) \left[\begin{array}{ll}6 & -0\end{array}\right]\left[\begin{array}{l}4 \\ 0\end{array}\right]

MDCAT/ ECAT question bank