2Midterm Part 2
Full Mark: 18
(1) Let A be the cofficenit matrix of the following system of equations:
Time 30 Minutes
[1+4=5 marks]
x1 + x2 = 3
-3x1 + 2x3 = 0
x2 -2x3 = 2
a) Write the matrix A.
b) Find the Adj. (A)
1 −3 0
9
2) Let 𝐴𝐴 = �−4 0
2 � and 𝑏𝑏 = �−6�. Define a transformation 𝑇𝑇: ℝ3 → ℝ3 by T(x)= Ax. If possible, find a
4
1 −2
4
vector x whose image under T is b. Otherwise, state that b is not in the range of the transformation T.
[5 marks]
3) Assume that T is a linear transformation. Find the standard matrix of 𝑇𝑇: ℝ2 → ℝ2 is a transformation that
maps 𝑒𝑒1 into 𝑒𝑒1 + 𝑒𝑒2 and maps 𝑒𝑒2 into 𝑒𝑒2 + 3𝑒𝑒1.
[4 marks]
1 −𝑘𝑘
0 1
4) Let A = �
0 0
0 1
0 0
−𝑘𝑘 0 �.
For what values of 𝑘𝑘 is the matrix A not invertible (singular)? [4 marks]
1 −𝑘𝑘
0 −1
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List of helpful Formula:
(1) Adj. A = Transpose of cofactor matrix of A;
(2) 𝐴𝐴−1 =
𝐴𝐴𝐴𝐴𝐴𝐴.𝐴𝐴
det 𝐴𝐴
(3) If 𝐴𝐴𝐴𝐴 = 𝑏𝑏 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑥𝑥 = 𝐴𝐴−1 𝑏𝑏
1