A matrix is a rectangular arrangement of numbers in rows and columns. It is used to represent data or solve mathematical problems. Matrices are fundamental in linear algebra with operations such as addition multiplication and inversion. They have applications in physics engineering computer science and economics.
Class 12 Maths Chapter 3 on Determinants covers square arrays of numbers used to solve systems of linear equations. It includes topics like properties of determinants calculation of 2×2 and 3×3 determinants cofactor expansion adjoint method Cramer’s rule and their applications in solving equations and finding areas of triangles using matrices.
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We are given a 3×3 matrix A that satisfies the equation:
A² = 4A – 3I
Step 1: Expressing A⁻¹
To find A⁻¹, we rearrange the given equation:
A² – 4A + 3I = 0
Factoring,
(A – I)(A – 3I) = 0
This implies that A satisfies the equation:
(A – I)(A – 3I) = 0
Multiplying both sides by (A – I)⁻¹ (if it exists),
A – 3I = (A – I)⁻¹ 0
Since A – I is invertible, we take its inverse on both sides,
A⁻¹ = 1/3 (4I – A)
Step 2: Selecting the Correct Option
Comparing with the given choices, the correct answer is: 1/3 (4I – A)
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