If |A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is
A square matrix is a matrix with an equal number of rows and columns. The order of a square matrix refers to its size, represented as “n x n,” where “n” is the number of rows (or columns). Examples include 2×2, 3×3, 4×4 matrices, and so on.
Class 12 Maths Chapter 4 focuses on Determinants. It covers properties of determinants such as addition subtraction multiplication of determinants along with cofactor expansion method. The chapter also includes applications like finding the area of a triangle solving systems of linear equations using Cramer’s rule and understanding the inverse of a matrix. This is crucial for CBSE Exam 2024-25.
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We are given that A is a square matrix of order 2, and |A| = |kA|. We have to compute the sum of all possible values of k.
First recall that if A is any square matrix of order 2, then the determinant of kA where k is any scalar is given by
|kA| = k² |A|
This is because for a 2×2 matrix, multiplication of the matrix A by scalar k scales the determinant by k².
We know that |A| = |kA|. Using the above formula, we substitute for the value of |kA|,
|A| = k² |A|
If |A| ≠ 0, we can divide both sides by |A| to get,
1 = k²
This gives two values for k
k = 1 or k = -1
Therefore, the sum of all possible values of k is:
1 + (-1) = 0
So, the correct answer is 0.
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