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2x2 Determinants - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Determinants are useful properties of square matrices, but can involve a lot of computation. A 2x2 determinant is much easier to compute than the determinants of larger matrices, like 3x3 matrices. To find a 2x2 determinant we use a simple formula that uses the entries of the 2x2 matrix. 2x2 determinants can be used to find the area of a parallelogram and to determine invertibility of a 2x2 matrix.

I want to talk about the determinant of a square matrix. Now every square matrix has a number associated which we call the determinant. For a 2 by 2 matrix, let's say the matrix is a, b, c, d we write the determinant two ways either determinant a like this or with vertical lines almost like absolute value but it doesn't mean absolute value in the context of of matrices this means the determinant and here's how we compute it.
We multiply diagonally a times d minus b times c so this diagonal minus this diagonal. Let's use that in a couple of examples, so let's calculate this determinant and you know it's a determinant when you see these vertical lines. It's going to be 5 times 6 or 30 minus 2 times 9 or 18 so that's going to be 12. Now this is also a determinant we calculate it exactly the same way 3 times 11 33 minus -5 times 4 -20 so that's 33+20 53 and now let's calculate this one -5 times 4 that's -20 minus 3 times 11 33 -53.
Something you should notice about determinants is if you switch if you switch two columns the determinant will turn out to be the opposite of what it originally was so switching to columns or switching to rows either way will change the sign of the determinant.

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