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Question
If A, B and C are angles of a triangle, then the determinant `|(-1, cos"C", cos"B"),(cos"C", -1, cos"A"),(cos"B", cos"A", -1)|` is equal to ______.
Options
0
– 1
1
None of these
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Solution
If A, B and C are angles of a triangle, then the determinant `|(-1, cos"C", cos"B"),(cos"C", -1, cos"A"),(cos"B", cos"A", -1)|` is equal to 0.
Explanation:
Let Δ = `|(-1, cos"C", cos"B"),(cos"C", -1, cos"A"),(cos"B", cos"A", -1)|`
C1 → aC1 + bC2 + cC3
⇒ `|(-"a" + "b" cos"C" + "c" cos "B", cos "C", cos"B"),("a" cos "C" - "b" + "c" cos"A", -1, cos"A"),("a"cos"b" + "b" cos"A" - "C", cos"A", -1)|`
⇒ `|(-"a" + "a", cos"C", cos"B"),(-"b" + "b", -1, cos"A"),(-"c" + "c", cos"A", -1)|` ....`[(because "From projection formula"),("a" = "b" cos"C" + "c" cos"B"),("b" = "a" cos "C" + "c" cos "a"),("c" = "b" cos "A" + "a" cos "B")]`
⇒ `[(0, cos "C", cos "B"),(0, -1, cos"A"),(0, cos"A", -1)]` = 0
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