|A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is
The sum of all possible values refers to adding together every possible outcome or result from a given set or situation. It involves considering all possible values and calculating their total. This concept is commonly used in problems involving probabilities or when determining the total of multiple solutions.
Chapter 4 of Class 12 Maths deals with Determinants. It explains the calculation of determinants using cofactor expansion and discusses properties. The chapter also covers applications like solving linear equations using Cramer’s rule and finding the inverse of matrices. Understanding these concepts is vital for success in the CBSE Exam 2024-25.
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To solve this, let’s break it down step by step.
We are given the equation |A| = |kA|, where A is a square matrix of order 2. That means it is a 2×2 matrix. We are trying to find the sum of all possible values of k.
Key Concepts:
1. Determinant of a matrix: For a square matrix A, the determinant is denoted as |A|.
2. Scalar multiplication and determinant:** Let A be any square matrix of order n and k be any scalar, then |kA| = k^n |A|.
Here:
– A is of the order of 2. Thus, n = 2.
– The determinant |kA| is defined as:
|kA| = k² |A|
Now, from |A|=|kA| we have:
|A| = k² |A|
If |A| ≠ 0, then divide both sides by |A|. Then
1 = k²
Hence, k = ±1
If |A| = 0, the equation holds true for all value of k.
Therefore, sum of all the possible values of k
= 1 + (-1)
= 0
Hence, sum of all possible values of k is zero.
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