To determine the matrix A, we need to solve the equation: A(
−1
2
3
1
)=(
−4
1
7
7
) Let's denote the elements of matrix A as: A=(
a
b
c
d
) Then we have the equation: (
a
b
c
d
)(
−1
2
3
1
)=(
−4
1
7
7
) Now perform the matrix multiplication on the left-hand side:
(
a
b
c
d
)(
−1
2
3
1
)=(
a(−1)+b(3)
a(2)+b(1)
c(−1)+d(3)
c(2)+d(1)
)
This simplifies to:
(
−a+3b
2a+b
−c+3d
2c+d
)=(
−4
1
7
7
)
We now have a system of linear equations: 1. −a+3b=−4 2. 2a+b=1 3. −c+3d=7 4. 2c+d=7 First, solve equations 1 and 2 for a and b. From equation 2 : 2a+b=1⟹b=1−2a Substitute b=1−2a into equation 1 : ‌−a+3(1−2a)=−4 ‌−a+3−6a=−4 ‌−7a+3=−4 ‌−7a=−7 ‌a=1 Then, substituting a=1 back into b=1−2a : b=1−2(1)=−1 So, a=1 and b=−1. Next, solve equations 3 and 4 for c and d. From equation 4: 2c+d=7⟹d=7−2c Substitute d=7−2c into equation 3 : ‌−c+3(7−2c)=7 ‌−c+21−6c=7 ‌−7c+21=7 ‌−7c=−14 ‌c=2 Then, substituting c=2 back into d=7−2c : d=7−2(2)=3 So, c=2 and d=3. Thus, the matrix A is: A=(