matrices worksheet ii
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8/7/2019 MATRICES WORKSHEET II
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I. LetM= , where a .
(a) FindM2 in terms ofa.
(b) IfM2 is equal to , find the value ofa.
(c) Using this value ofa, find Ml and hence solve the system of equations:
x + 2y = 3
2x y = 3
II. A and B are 2 2 matrices, where A = and BA = . Find B.III. Given the matrix A = find the values of the real number kfor which
det(A kI) = 0 where I = .
IV. (a) Find the values ofa and b given that the matrixA =is the inverse of the matrixB =
(b) For the values ofa and b found in part (a), solve the system of
linear equations
x + 2y2z= 5
x+ y 3z= a 1.
3x + by + z = 0
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a
5
4
4
5
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0
5
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844
0
3
0
0
4
7
6
3
5
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5
8
a
.
3
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1
3
1
b
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V. Find the values of the real numberkfor which the determinant of the matrixis equal to zero.
VI.IfA = andB = , find 2 values ofx and y, given thatAB =BA.
VII. The matrix is singular. Find the values ofk.
VIII. The matrixA is given by
A =
Find the values ofkfor whichA is singular.
IX.The matricesA,B,Xare given byA= B= X= where a,b,c,d .
Given thatAX+X=, find the exact values ofa,b,c and d.
X.Given thatA = andI= , find the values of for which (AI) isa singular matrix.
XI.Consider the matrixA = .
XII. (a) Write down the inverse,Al.(b) B, CandXare also 2 2 matrices.
(i) Given thatXA +B = C, expressXin terms ofA1,B and C.
(ii) Given thatB = , and C= findX.
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2 k
k
2
x
48
2 y
k
k
53
131
321
243
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k
k
,65
13
,
30
84
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dc
ba
43
23
10
01
17
25
25
76,
78
05
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XIII. The matricesA,B, CandXare all non-singular 3 3 matrices.Given thatAlXB = C, expressXin terms of the other matrices.
XIV. Given that the matrixA= is singular, find the value ofp.XV. The square matrixXis such thatX3 = 0. Show that the inverse of the matrix (I
X) isI+X+X2.
XVI. Let C= andD = .The 2 2 matrix Q is such that 3Q = 2CD
(a) Find Q.
(b) Find CD.
(c) FindD1.
XVII. Given thatA = andB = findXifBX=AAB.
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211
p
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a
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5
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