To find matrix B, we first express the given equations as: ‌2A+3B=[
2
−1
4
3
2
5
], ‌A+2B=[
5
0
3
1
6
2
]. We can rearrange these equations to solve for A and B. Let's try to eliminate A by multiplying the second equation by 2 and subtracting it from the first equation: Multiply A+2B=[
5
0
3
1
6
2
] by 2 : 2(A+2B)=2[
5
0
3
1
6
2
]=[
10
0
6
2
12
4
] Now subtract this from the first equation:
‌2A+3B−(2(A+2B))=[
2
−1
4
3
2
5
]−[
10
0
6
2
12
4
] ‌2A+3B−2A−4B=[
2−10
−1−0
4−6
3−2
2−12
5−4
]
‌−B=[
−8
−1
−2
1
−10
1
] Multiplying both sides by -1 , we get: B=[
8
1
2
−1
10
−1
] Comparing this matrix B with the options given: Option C is: [
8
1
2
−1
10
−1
] Clearly, option C matches our calculation for B. Hence, the correct answer is Option C.