Problem1:Find the augmented matrix for the following system of linear equations |
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v + 2 w - y + z = 1 3 w + x - z = 2 x + 7 y = 1
Solution1
Problem2:Find the system of linear equations corresponding to the augmented matrix |
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[3 0 -2 5] [ ] [7 1 4 -3] [ ] [0 -2 1 7]
Solution2
Problem3:The curve, |
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2 y = a x + b x + c
passes through the points (x1,y1), (x2,y2), (x3,y3). Show that the coefficients a,b, and c are a solution of the system of linear equations whose augmented matrix is, |
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[ 2 ] [x1 x1 1 y1] [ ] [ 2 ] [x2 x2 1 y2] [ ] [ 2 ] [x3 x3 1 y3]
Solution3
Problem4:Find the solution to the system of equations with augmented matrix given by, |
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[1 0 0 -7 8] [ ] [0 1 0 3 2] [ ] [0 0 1 1 -5]
Solution4
Problem5:Use Gauss-Jordan elimination to find the solution to: |
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5 x - 2 y + 6 z = 0 -2 x + y + 3 z = 1
Solution5
Problem6:Find coefficients a,b,c, and d so that the curve, |
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2 2 a x + a y + b x + c y + d = 0
goes through the points: (-4,5), (-2,7) and (4,-3)
Plot this curve.
Solution6
|
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[cos(t) sin(t)] [ ] [-sin(t) cos(t)]
Solution7
Problem8:Let A, B, and 0 be 2x2 matrices. Assuming that A is invertible, find a matrix C so that, |
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[ -1 ] [A 0 ] [ ] [ -1] [ C A ]
is the inverse of the partitioned matrix |
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[A 0] [ ] [B A]
Solution8
Problem9:Use the result of the previous problem to find the inverse of the matrix, |
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[ 1 1 0 0] [ ] [-1 1 0 0] [ ] [ 1 1 1 1] [ ] [ 1 1 -1 1]
Solution9
Problem10:Which of the following are elementary matrices? |
> (a);
[ 1 0] [ ] [-5 1]> (b);
[0 0 1] [ ] [0 1 0] [ ] [1 0 0]> (c);
[2 0 0 2] [ ] [0 1 0 0] [ ] [0 0 1 0] [ ] [0 0 0 1]
Solution10
Problem11:Consider the matrices, |
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[3 4 1] [ ] A := [2 -7 -1] [ ] [8 1 5]> ;
[3 4 1] [ ] B := [2 -7 -1] [ ] [2 -7 -3]
Find elementary matrices E and F such that, |
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E A = B F B = A
Solution11
Problem12:Find the inverse of each of the following matrices, where a,b,c,d, and e are all nonzero, |
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[a 0 0 0] [ ] [0 b 0 0] A := [ ] [0 0 c 0] [ ] [0 0 0 d] [0 0 0 a] [ ] [0 0 b 0] B := [ ] [0 c 0 0] [ ] [d 0 0 0] [e 0 0 0] [ ] [1 e 0 0] C := [ ] [0 1 e 0] [ ] [0 0 1 e]
Solution12
Problem13:Find the conditions that b's must satisfy for the system to be consistent |
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x - 2 y - z = b1 -4 x + 5 y + 2 z = b2 -4 x + 7 y + 4 z = b3
Solution13
Problem14:Consider the matrices, |
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[2 1 2] [ ] A := [2 2 -2] [ ] [3 1 1] [x1] [ ] x := [x2] [ ] [x3]
Show that the equation Ax=x can be rewritten as (A-I)x=0 and use this result to solve Ax=x for x. Also solve Ax=4x. |
Solution14
Problem15:Let A be a symmetric matrix.
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Solution15
Problem16:A square matrix A is called skew-symmetric if, |
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T A = -A
Show that
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Solution16
Problem17:Find an upper triangular matrix A, such that, |
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3 [1 30] A = [ ] [0 -8]
Solution17 |