Find the multiplicative inverse of each matrix, if it exists...

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied by the original matrix, yields the identity matrix. This process is only possible for square matrices that are nonsingular, meaning they have a non-zero determinant. The concept is central in solving systems of linear equations and understanding linear transformations.

Determinant

The determinant is a scalar value that provides important properties of a matrix, such as its invertibility. A matrix is invertible if and only if its determinant is non-zero. The calculation of the determinant can involve various techniques, including cofactor expansion or more efficient algorithms for larger matrices, and it is fundamental in linear algebra.

Elementary Row Operations and Gauss-Jordan Elimination

Elementary row operations, which include row swapping, scaling, and row addition, are used in the Gauss-Jordan elimination method to transform a matrix into its reduced row echelon form. This method is a systematic procedure for finding the matrix inverse by augmenting the original matrix with the identity matrix and then performing operations until the identity matrix is obtained on the left side. This approach is particularly useful for larger matrices.

Adjugate (or Adjoint) Matrix

The adjugate of a matrix, also known as the adjoint matrix, is the transpose of the cofactor matrix. It plays a key role in the formula for the inverse of a matrix, where the inverse is given by the adjugate divided by the determinant. Understanding the adjugate matrix is essential to computing inverses using the classical method.

Transcript

00:01 So we want to find the multiplicative inverse of the following matrix.

00:05 And to do so, we're going to augment it with the inverse.

00:10 So the resulting matrix will be 1, 2, negative 1, negative 3, 4 and 1, negative 2, negative 4, negative 5.

00:21 And the identity matrix for 3 by 3 is just 1, 0 ,0, 0 ,01.

00:30 And by using row operations, we want to transform the left -hand side of the matrix into the right -hand side.

00:38 And in doing so, this left -hand side will give us the inverse.

00:43 In other words, we want to transform a into i so that i becomes a -inverse.

00:52 So the first step to do so will be to switch r3 with r2.

01:01 So we'll have 1, 2, negative 1, negative 2, sorry, negative 2, negative 4, negative 5, negative 3, 4, and 1.

01:20 And the right hand side will have 1, 0 ,0, 0 ,01, and 010.

01:28 Now we're going to multiply r1 by negative 5 so that we can add it to r2.

01:37 To find any new r2, so the first and third row, the same the second row just becomes negative 7 negative 14 and 0 and this constant right here just becomes negative 5 0 and 1 now the second the following step will be to multiply r1 by 3 and added to r3 so we'll have 1 to negative 1 negative 7 negative 0 0 10 and negative 2 on the right hand side will have 1 0 -0 -0 -0 -0 -1 3 -0 then we're going to multiply r3 by negative 1 1 half and we're going to add it to r1 so r1 just becomes 1 negative 3 0 negative 1 1 1 1 1 1 1 0 0 and the rest is the same negative 7 negative 7 negative 14 0, negative 5, 0 and 1, 0, 10, negative 2, 3, 1, 0.

03:23 Now we're going to multiply r1 by 7 to get rid of this 7 over here, and we're going to add it to our 2 to do so.

03:34 So we'll have 1, negative 3, 0, 010, negative 2, negative 1 over 2, negative 1 over 2, 0 ,0, 310, and the 2.

03:49 Second row just becomes 0, negative 35, 0, negative 16 divided by 2, and negative 7 divided by 2, sorry, and 1...

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