Worksheet featuring practice problems on matrices and simultaneous equations, covering topics like determinants, matrix inverses, and solving linear systems.

1. a. det(A) = (3)(-3) - (2)(2) = -9 - 4 = -13
b. A⁻¹ = (1/-13) * [ -3 -2 ]
[ -2 3 ] = [ 3/13 2/13 ]
[ 2/13 -3/13 ]
c. [x] = A⁻¹ * [1] = [ 3/13 2/13 ] * [1] = [ (3/13)(1) + (2/13)(5) ] = [ 13/13 ] = [1]
[y] [5] [ 2/13 -3/13 ] [5] [ (2/13)(1) + (-3/13)(5) ] [-13/13] [-1]

2. a. [1 -1][x] = [-5]
[3 2][y] [-5]
b. Let B = [1 -1]. det(B) = (1)(2) - (-1)(3) = 2 + 3 = 5.
[3 2]
B⁻¹ = (1/5) * [ 2 1] = [ 2/5 1/5]
[-3 1] [-3/5 1/5]
c. [x] = B⁻¹ * [-5] = [ 2/5 1/5] * [-5] = [ (2/5)(-5) + (1/5)(-5) ] = [ -10/5 - 5/5 ] = [ -15/5 ] = [-3]
[y] [-5] [-3/5 1/5] [-5] [ (-3/5)(-5) + (1/5)(-5) ] [ 15/5 - 5/5 ] [ 10/5 ] [ 2]

3. a. Let C = [1 4]. det(C) = (1)(1) - (4)(2) = 1 - 8 = -7.
[2 1]
C⁻¹ = (1/-7) * [ 1 -4] = [ -1/7 4/7]
[-2 1] [ 2/7 -1/7]
b. From C * [a b] = [4 13], we have [a b] = C⁻¹ * [4 13].
[c d] [8 5] [c d] [8 5]
[a b] = [ -1/7 4/7] * [4 13] = [ (-1/7)(4) + (4/7)(8) (-1/7)(13) + (4/7)(5) ] = [ (-4+32)/7 (-13+20)/7 ] = [28/7 7/7] = [4 1]
[c d] [ 2/7 -1/7] [8 5] [ (2/7)(4) + (-1/7)(8) (2/7)(13) + (-1/7)(5) ] [ (8-8)/7 (26-5)/7 ] [ 0/7 21/7] [0 3]
So, a=4, b=1, c=0, d=3.

4. a. P = [2 1]. det(P) = (2)(-3) - (1)(5) = -6 - 5 = -11.
[5 -3]
P⁻¹ = (1/-11) * [ -3 -1] = [ 3/11 1/11]
[ -5 2] [ 5/11 -2/11]
b. The equation x[2] + y[1] = [7] is equivalent to P*[x] = [7].
[5] [-3] [1] [y] [1]
So, [x] = P⁻¹ * [7] = [ 3/11 1/11] * [7] = [ (3/11)(7) + (1/11)(1) ] = [ (21+1)/11 ] = [22/11] = [2]
[y] [1] [ 5/11 -2/11] [1] [ (5/11)(7) + (-2/11)(1) ] [ (35-2)/11 ] [33/11] [3]

5. a. M = [2 5]. det(M) = (2)(15) - (5)(7) = 30 - 35 = -5 ≠ 0, so M is non-singular.
[7 15]
b. M⁻¹ = (1/-5) * [ 15 -5] = [ -3 1]
[ -7 2] [ 7/5 -2/5]
c. M * M⁻¹ = I = [1 0]
[0 1]
d. Pre-multiplying both sides of M*[x] = [-3] by M⁻¹ gives:
[y] [17]
M⁻¹ * M * [x] = M⁻¹ * [-3]
[y] [17]
I * [x] = M⁻¹ * [-3]
[y] [17]
e. [x] = M⁻¹ * [-3] = [ -3 1] * [-3] = [ (-3)(-3) + (1)(17) ] = [ 9 + 17 ] = [26]
[y] [17] [ 7/5 -2/5] [17] [ (7/5)(-3) + (-2/5)(17) ] [ (-21 - 34)/5 ] [-55/5] = [-11]
So, x = 26, y = -11.