Matrices and Their Inverses — Quiz

Question 2 • Advanced Algebra: Concepts and Connections

Which of the following represents the inverse of the matrix? cap A is equal to the 2 by 2 matrix row 1, A b row 2, c d

Answer

the fraction with numerator 1 and denominator absolute value of cap A the 2 by 2 matrix, row 1, column 1, negative d column 2, b row 2, column 1, c column 2, negative A

the fraction with numerator 1 and denominator absolute value of cap A the 2 by 2 matrix, row 1, column 1, d column 2, negative c row 2, column 1, negative b column 2, A

the fraction with numerator 1 and denominator absolute value of cap A the 2 by 2 matrix, row 1, column 1, A column 2, negative b row 2, column 1, negative c column 2, d

the fraction with numerator 1 and denominator absolute value of cap A the 2 by 2 matrix, row 1, column 1, d column 2, negative b row 2, column 1, negative c column 2, A

Explanation

For a 2 × 2 matrix A = [[a, b], [c, d]], its inverse is (1 / det(A)) × [[d, −b], [−c, a]], provided det(A) = ad − bc is nonzero. This swaps the diagonal entries and changes the signs of the off-diagonal entries. Choice D gives exactly that matrix.

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