Video: Evaluating Determinants

Find the value of |π‘Ž + π‘₯, 5π‘Ž and 𝑏 + 7𝑦, βˆ’7𝑏|.

01:46

Video Transcript

Find the value of the determinant of the matrix π‘Ž plus π‘₯, five π‘Ž, 𝑏 plus seven 𝑦, negative seven 𝑏.

Remember, for a two-by-two matrix 𝐴 with elements π‘Ž, 𝑏, 𝑐, 𝑑, its determinant can be found by subtracting the product of elements 𝑏 and 𝑐 from the product of elements π‘Ž and 𝑑. In our matrix, the element π‘Ž is given as π‘Ž plus π‘₯. 𝑏 is given as five π‘Ž. 𝑐 is 𝑏 plus seven 𝑦. And 𝑑 is negative seven 𝑏.

To find its determinant then, we need to find the product of π‘Ž and 𝑑. That’s π‘Ž plus π‘₯ multiplied by negative seven 𝑏. I inverted that as negative seven 𝑏 multiplied by π‘Ž plus π‘₯. We’re then going to subtract the product of the elements in the top right and bottom left. That’s 𝑏 and 𝑐. And we’re going to subtract five π‘Ž multiplied by 𝑏 plus seven 𝑦.

We’ll need to expand each of these brackets as normal. Negative seven 𝑏 multiplied by π‘Ž is negative seven π‘Žπ‘. And negative seven 𝑏 multiplied by π‘₯ is negative seven 𝑏π‘₯. Negative five π‘Ž multiplied by 𝑏 is negative five π‘Žπ‘. And negative five π‘Ž multiplied by seven 𝑦 is negative 35π‘Žπ‘¦.

Finally, we need to simplify this expression by collecting like terms. Here, we have two terms that are some multiple of π‘Žπ‘. Negative seven π‘Žπ‘ minus five π‘Žπ‘ is negative 12π‘Žπ‘.

And therefore, the determinant of our matrix is negative seven 𝑏π‘₯ minus 35π‘Žπ‘¦ minus 12π‘Žπ‘.

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