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Question
The sum of the products of elements of any row with the co-factors of corresponding elements is equal to ______.
Solution
The sum of the products of elements of any row with the co-factors of corresponding elements is equal to the value of the determinant of the given matrix.
Explanation:
Let Δ = `|("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)|`
Expanding along R1
`"a"_11 |("a"_22, "a"_23),("a"_32, "a"_33)| - "a"_12 |("a"_21, "a"_23),("a"_31, "a"_33)| + "a"_13 |("a"_21,"a"_22),("a"_31, "a"_32)|`
⇒ `"a"_11"M"_11 + "a"_12"M"_12 + "a"_13"M"_13` ....(Where M11, M12 and M13 are the minors of the corresponding elements)
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