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Question Video: Solving Determinants Using Properties Mathematics

Consider [𝑥, 𝑦 and 𝑧, 𝑤] = 6. Find the value of [(𝑥 − 10𝑦), 𝑦 and (𝑧 − 10𝑤), 𝑤].

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Video Transcript

Consider the determinant of the two-by-two matrix 𝑥, 𝑦, 𝑧, 𝑤 is equal to six. Find the value of the determinant of the two-by-two matrix 𝑥 minus 10𝑦, 𝑦, 𝑧 minus 10𝑤, 𝑤.

In this question, we’re given the determinant of a two-by-two matrix and asked to determine the determinant of a different two-by-two matrix. And both of these matrices contain four unknowns, so we can’t evaluate these directly. We might be tempted to expand the determinant in our first matrix and then expand the determinant in our second matrix and try to rewrite this expression in terms of our first determinant. However, there’s a much simpler method involving the properties of determinants.

To do this, we need to notice something interesting about the second matrix we’re given. In the first entry of the first column, we’re subtracting 10𝑦, and in the second entry of the second column, we’re subtracting 10𝑤. This is a scalar multiple of the second column of our first matrix. In fact, we’re subtracting this directly from the first column of this matrix. In other words, to generate the second matrix, we’re subtracting 10 lots of the second column from the first column of our first matrix. And we can recall adding and subtracting scalar multiples of one row or column to a different row or column will not affect the determinant. Therefore, the determinant of the second matrix will be equal to the determinant of the first matrix, which we’re told is equal to six.

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