In Exercises 13-18, use the fact that if A=[ a b c d ], then A^-1=(1)/(a d-b c)[

In Exercises 13-18, use the fact that if A=[ a b c...

In Exercises $13-18,$ use the fact that if $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right],$ then $A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \\ {-c} & {a}\end{array}\right]$ to find the inverse of each matrix, if possible. Check that $A A^{-1}=I_{2}$ and $A^{-1} A=I_{2}$ $$ A=\left[\begin{array}{ll}{2} & {-6} \\ {1} & {-2}\end{array}\right] $$

In Exercises $13-18,$ use the fact that if $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right],$ then $A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \\ {-c} & {a}\end{array}\right]$ to find the inverse of each matrix, if possible. Check that $A A^{-1}=I_{2}$ and $A^{-1} A=I_{2}$
$$
A=\left[\begin{array}{ll}{2} & {-6} \\ {1} & {-2}\end{array}\right]
$$

Key Concepts

Inverse of a Matrix

The inverse of a matrix is a matrix which, when multiplied with the original, yields the identity matrix. It is fundamental in solving systems of linear equations and understanding linear transformations, as the inverse effectively 'undoes' the effect of the original matrix.

Determinant

The determinant is a scalar value computed from a square matrix’s elements that provides important information about the matrix. For a 2x2 matrix, the determinant is calculated as ad - bc, and it indicates whether the matrix is invertible; a nonzero determinant means the matrix has an inverse.

Invertibility Conditions

A matrix is invertible (or non-singular) if its determinant is nonzero. This condition ensures that there exists a unique matrix which, when multiplied with the original, results in the identity matrix. If the determinant is zero, the matrix does not have an inverse.

Matrix Multiplication and Identity Matrix

Matrix multiplication is used to combine matrices, and the identity matrix serves as the multiplicative identity in matrix algebra. Confirming that the product of a matrix and its inverse equals the identity matrix is crucial, as it verifies the correctness of the inverse calculation.

Transcript

00:01 Okay, so in this problem, we're given this matrix a here, which is 2 -6, 1 -2, and we need to find a inverse using the special formula that we're given.

00:11 And then we need to verify that a -inverse is correct by multiplying a times an a -inverse and a inverse times a.

00:17 And in both of those scenarios, we should get the identity matrix back.

00:21 And so we're going to do this step -by -step.

00:23 So to get a -inverse, there's a special formula.

00:26 So it's going to be 1 over ad, which is negative 2 times 2, so that's negative 4, minus bc, which is negative 6 times 1.

00:38 So it's going to be minus negative 6, in other words, plus 6.

00:43 And then we're going to multiply that fraction, whatever it comes out to be, by this new matrix, where we flip the things on this diagonal, and we negate the things on this diagonal.

00:56 So if we simplify this matrix out, we multiply this fraction, which negative 4 plus 6 here is going to give us just positive 2.

01:08 So that's going to be 1 half times each of these entries in this matrix.

01:12 So it's going to be 1 half times negative 2, which is negative 1.

01:15 It means 1⁄2, which is 3.

01:18 It's going to be 1⁄2 times negative 1⁄2, which is negative 1⁄2.

01:22 And it's going to be 1⁄2 times 2, which is 1.

01:26 So so far, this is looking like our candidate for a inverse.

01:29 We need to just verify that it works by multiplying it by a, and by multiplying a by it, in order to see if it gives us the correct results.

01:39 So to do that, we're going to take a, which has a negative 2 right here, and we're going to multiply that by a inverse.

01:49 And multiplying this, we should get the 2x2 identity matrix back.

01:51 So let's see if that happens.

01:54 Take the first row, multiply it by the first column here, see what we get.

01:58 It's going to be 2 times negative 1, which is negative 2.

02:02 And then we're going to add that to negative 6 times a half times negative 1, which is 3.

02:07 So it's negative 2 plus 3, which is 1.

02:10 So we're good there.

02:12 Multiply that first row times the second column.

02:14 It's going to be 2 times 3, which is 6 plus negative 6 times 1, which is negative 6.

02:19 So that's 6 minus 6, which is 0.

02:22 So we're good there.

02:23 Now we're going to take the second row and do the same thing...

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