Question 18 • Advanced Algebra: Concepts and Connections
If cap A is equal to the 2 by 2 matrix, row 1, column 1, 4 column 2, 2 row 2, column 1, negative 7 column 2, negative 3 and cap b is equal to the 2 by 2 matrix, row 1, column 1, negative 3 column 2, 4 row 2, column 1, negative 1 column 2, 5 , what is the solution to the equation cap A cap x is equal to cap b ?
the 2 by 2 matrix, row 1, column 1, 18 and 1 half column 2, 11 row 2, column 1, 19 column 2, 11
C
the 2 by 2 matrix, row 1, column 1, 5 and 1 half column 2, negative 11 row 2, column 1, negative 12 and 1 half column 2, 24
D
the 2 by 2 matrix, row 1, column 1, negative 3 halves column 2, 2 row 2, column 1, 1 seventh column 2, negative 1 and 2 thirds
Explanation
To solve AX = B, multiply both sides by A⁻¹. The determinant of A is 4(−3) − 2(−7) = 2, so A⁻¹ = ½[[-3, −2], [7, 4]]. Then A⁻¹B = [[11/2, −11], [−25/2, 24]], which is [[5½, −11], [−12½, 24]]. This matches choice C.
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