MATH 122 โ H Activity 2
Linear Algebra and Matrix Theory Regine Joyce A. Camacho โ BS Mathematics
I. MODIFIED TRUE OR FALSE.
1. FALSE. Type 3
2. FALSE. Type 2
3. FALSE. Only trivial solution
4. TRUE
5. FALSE. Has no zero row.
II. COMPUTATIONS.
1. Let ๐ด=[2 โ1
3 2] and ๐ต=[โ4 2
1 3]. Using these matrices, show that
a. (๐ด+๐ต)2โ ๐ด2+2๐ด๐ต+๐ต2
Solution:
Given that ๐ด=[2 โ1
3 2] and ๐ต=[โ4 2
1 3], ๐ด+๐ต is
[2 โ1
3 2]+[โ4 2
1 3]=[โ2 1
4 5]
With that, (๐ด+๐ต)2=(๐ด+๐ต)(๐ด+๐ต) is
[โ2 1
4 5][โ2 1
4 5]=[(โ2)(โ2)+(1)(4) (โ2)(1)+(1)(5)
(4)(โ2)+(5)(4) (4)(1)+(5)(5)]
=[4+4 โ2+5
โ8+20 4+25]
โ(๐ด+๐ต)2=[8 3
12 29]
Since ๐ด=[2 โ1
3 2], this implies that ๐ด2=(๐ด)(๐ด) is
[2 โ1
3 2][2 โ1
3 2]=[(2)(2)+(โ1)(3) (2)(โ1)+(โ1)(2)
(3)(2)+(2)(3) (3)(โ1)+(2)(2)]
=[4โ3 โ2โ2
6+6 โ3+4]
โ๐ด2=[1 โ4
12 1]
Since ๐ด=[2 โ1
3 2] and ๐ต=[โ4 2
1 3], this implies that ๐ด๐ต is
[2 โ1
3 2][โ4 2
1 3]=[(2)(โ4)+(โ1)(1) (2)(2)+(โ1)(3)
(3)(โ4)+(2)(1) (3)(2)+(2)(3)]
=[โ8โ1 4โ3
โ12+2 6+6]
โ๐ด๐ต=[โ9 1
โ10 12]
With that, 2๐ด๐ต is 2[โ9 1
โ10 12]=[โ18 2
โ20 24]
