In Problems 31-40, each matrix is nonsingular. Find the inverse of each matrix. [ 2 1

In Problems 31-40, each matrix is nonsingular. Find the...

Key Concepts

2×2 Matrix Inversion Formula

For a 2×2 matrix, the inverse can be computed using a specific formula where the positions of the a and d elements are swapped, the signs of the b and c elements are reversed, and the entire matrix is multiplied by the reciprocal of the determinant. This concise formula simplifies the process of finding the inverse in lower-dimensional cases.

Nonsingular Matrix

A nonsingular matrix is one that has a nonzero determinant, which guarantees that an inverse exists for the matrix. This property is essential in ensuring that solutions to the associated system of linear equations are unique.

Determinant

The determinant is a scalar value that summarizes certain properties of a square matrix, including whether the matrix is singular or nonsingular. For a 2×2 matrix, the determinant is calculated as ad - bc, and it must be nonzero for the matrix to have an inverse.

Matrix Inversion

Matrix inversion is the process of finding a matrix, called the inverse, such that when it is multiplied by the original matrix, the result is the identity matrix. This concept is fundamental in solving linear systems, transforming coordinates, and many areas of applied mathematics.

Transcript

00:01 Hello, so we're given our matrix a, a matrix capital a, we're told, which is 2 .1aa, so the a is not equal to 0.

00:09 And we want to find the inverse matrix.

00:13 So we just take our matrix a, we put this vertical line, and then we concatenated here with the, well, the 2x2 identity matrix, which is just a 2x2 matrix, which has ones down the main diagonal and 0s everywhere else.

00:24 Now we're going to perform just elementary row operations to put this in row echelon form, i'm basically putting the identity we want on the left -hand side, and then what we end up with on the right -hand side of the vertical bar is going to be our inverse matrix.

00:38 So what do we want? we have a two now in our leading term, in our first row, first column.

00:45 We want that to be a one.

00:46 So let's just do one -half times our row one.

00:50 So our first thing we're going to do here is going to take row one, and say this is equal to one -half, 1 1 half times row 1 and then we want well a 0 0 beneath it right so then our next move is going to be row 2 equal to well row 2 minus a times row 1 and that is going to then put a 0 beneath it so performing that we end up with 1 1 1 half 0 0, and then we have a over 2, and then our vertical bar, and on the right side of that vertical bar, we then have 1 -half, 0, negative a over 2, and 1.

01:44 Okay, so then we need to, well, we have this a over 2 in our second row, second column, want that to be a 1, and they have a 0 above it.

01:52 So the operations we perform now is saying, well, row 2 is going to be 2.

02:00 Over a, right, 2 over a, again, just a constant...

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