Given that A = [aᵢⱼ] is a square matrix of order 3 x 3 and |A| = -7, then the value of ∑³₁ ₌ ᵢ aᵢ₂ A ᵢ₂, where Aᵢⱼdemotes the cofactor of element a ᵢⱼis:
The cofactor of an element in a matrix is the signed minor of that element. To find it, delete the row and column containing the element, then calculate the determinant of the remaining matrix. The cofactor is the minor multiplied by +1 or -1 depending on the position.
Chapter 4 of Class 12 Maths is about Determinants. It includes the calculation of determinants using methods like cofactor expansion and the properties of determinants. The chapter also covers the application of determinants in solving linear equations using Cramer’s rule and finding the inverse of matrices. These concepts are crucial for the CBSE Exam 2024-25.
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We are given that A is a square matrix of order 3×3 and |A| = -7. We are asked to find the value of the sum:
∑₁³ aᵢ₂ Aᵢ₂, where Aᵢ₂ denotes the cofactor of the element aᵢ₂.
This sum corresponds to the determinant of matrix A when expanding along the second column. Specifically, the cofactor expansion formula for the determinant of a matrix A is:
|A| = ∑ᵢ aᵢⱼ Aᵢⱼ
where aᵢⱼ is the element of the matrix and Aᵢⱼ is its cofactor. Here, the expansion is along the second column. Thus, the sum becomes,
∑₁³ aᵢ₂ Aᵢ₂ = |A|
As we know that |A| = -7, therefore the value of the sum is -7.
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