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Question
The determinant `|(sin"A", cos"A", sin"A" + cos"B"),(sin"B", cos"A", sin"B" + cos"B"),(sin"C", cos"A", sin"C" + cos"B")|` is equal to zero.
Options
True
False
Solution
This statement is True.
Explanation:
Let Δ = `|(sin"A", cos"A", sin"A" + cos"B"),(sin"B", cos"A", sin"B" + cos"B"),(sin"C", cos"A", sin"C" + cos"B")|`
Splitting up C3
= `|(sin"A", cos"A", cos"B"),(sin"B", cos"A", cos"B"),(sin"C", cos"A", cos"B")| + |(sin"A", cos"A", cos"B"),(sin"B", cos"A", cos"B"),(sin"C", cos"A", cos"B")|`
= `0 + |(sin"A", cos"A", cos"B"),(sin"B", cos"A", cos"B"),(sin"C", cos"A", cos"B")|` ....[∵ C1 and C3 are identical]
= `cos"A" cos"B" |(sin"A", 1, 1),(sin"B", 1, 1),(sin"C", 1, 1)|`
[Taking cos A and cos B common from C2 and C3 respectively]
= cos A cos B (0) ....[∵ C2 and C3 are identical]
= 0
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