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

step 1 In this problem, we want to find a inverse given this A matrix here. And we know that, actually,

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 $A^{-1} A=I_{2}$ $$ A=\left[\begin{array}{rl}{2} & {3} \\ {-1} & {2}\end{array}\right] $$

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 $A^{-1} A=I_{2}$
$$
A=\left[\begin{array}{rl}{2} & {3} \\ {-1} & {2}\end{array}\right]
$$

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix, known as the inverse, that when multiplied by the original matrix yields the identity matrix. It is a key concept in linear algebra, enabling the solution of systems of linear equations, transformation analysis, and more. For a square matrix, the inverse exists only if the matrix is non-singular (i.e., its determinant is non-zero).

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix, including whether the inverse exists. For a 2x2 matrix, the determinant is calculated as ad - bc. A non-zero determinant indicates that the matrix is invertible, while a zero determinant signifies that the matrix is singular and does not have an inverse.

Matrix Multiplication

Matrix multiplication is the operation of multiplying two matrices to produce a new matrix. In the context of matrix inversion, the product of a matrix and its inverse must yield the identity matrix. This verification ensures that the computed inverse is correct and that the operations adhere to the rules of linear algebra.

Identity Matrix

The identity matrix is a special kind of square matrix where all the elements on the main diagonal are one, and all other elements are zero. It acts as the multiplicative identity in matrix algebra, meaning any matrix multiplied by the identity matrix remains unchanged. This property is used to verify the correctness of an inverse matrix.

Transcript

00:01 In this problem, we want to find a inverse given this a matrix here.

00:07 And we know that, actually, i'll go back, if we have that some a matrix is a, b, c, d, just any numbers, a, b, c, and d, that a inverse is equal to 1 over a, d minus b, c, c, and d.

00:30 Times the matrix d minus b minus c.

00:39 All right, so we're going to use this formula to find the inverse of a here.

00:44 So we want to plug into our formula.

00:49 So a inverse equals.

00:53 Well, actually, it might be easier here to just note that in this case, if we put this a in form of our general matrix here, that little a equals two, little b.

01:05 Equals 3, little c equals negative 1, and d equals 2.

01:11 Now it'll be just a bit easier to plug into our formula.

01:15 All right, so we do 1 over a times d, so 2 times 2, minus b times c.

01:22 So negative 1 times 3, or 3 times negative 1.

01:28 All right, and we're multiplying that by this matrix on the inside, which is d, which is 2, negative, negative, be negative 3, negative c, negative negative 1, it's 1, and a, which is 2.

01:44 All right, we can simplify this a little bit to see that our first part here is equal to 1 over 4 plus 3, right? because 2 times 2 is 4, and then minus the negative is plus and 1 times 3 is 3.

02:02 All right, so 1 over 7.

02:05 So we want to multiply each of our entries by 1 over 7.

02:10 So we get 2 over 7, negative 3 over 7, 1 over 7, and 2 over 7.

02:22 All right.

02:23 So now we have our matrix, our inverse of a matrix.

02:28 Now we just want to double check that when we multiply a times a inverse, that we get the identity.

02:34 And when we multiply a inverse times a, we also get the identity.

02:41 So let's go ahead and do that.

02:43 Starting with a times a inverse, that will be 2, 3, negative 1, 2, multiply by a inverse, which we said is 2 7ths, negative 3, 7th, 1 7th, and 2 7ths.

03:08 So we get that that's equal to, we do the first row of a times the first column of a inverse.

03:19 2 times 2 7s is 4 7s plus 3 times 1 7th is 3 7s.

03:26 All right, now let's do the first row of a times the second column of a inverse...

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