A matrix is a rectangular array of numbers arranged in rows and columns. It is used to represent data or solve mathematical problems. Matrices are involved in various operations like addition multiplication and inversion. They are fundamental in linear algebra and applied in physics engineering and computer science.
Class 12 Maths Chapter 3 focuses on Determinants which are square arrays of numbers used to solve systems of linear equations. Topics include properties of determinants determinant of 2 x 2 and 3 x 3 matrices cofactor and adjoint methods and their applications in finding areas and solving linear equations using Cramer’s rule and matrix inversion. Determinants are crucial in advanced algebra and geometry.
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We are given that P is a 3 × 3 matrix such that P’ = 2P + I, where P’ is the transpose of P.
Step 1: Take the transpose of both sides
We take the transpose of both sides of the equation P’ = 2P + I:
(P’)’ = (2P + I)’
Since the transpose of the transpose of a matrix is the matrix itself, we get:
P = 2P’ + I
Step 2: Plug P’ = 2P + I into this equation
Next, plug the expression P’ = 2P + I into the equation:
P = 2(2P + I) + I
P = 4P + 2I + I
P = 4P + 3I
Step 3: Move terms around
Now we move the terms around in the equation.
P – 4P = 3I
-3P = 3I
P = -I
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