Show that if det[ a11 a12 a21 a22 ] α+det[ a11 a13 a21 a23 ] β+det

step 1 Alrighty, folks. So here we have three matrices, A1, A2, and A3, where A1 is equal to one -half

Show that if det[ a11 a12 a21 a22 ] α+det[ ...

Key Concepts

Determinant and Minors

The determinant is a scalar value computed from a square matrix that encodes important properties such as invertibility and linear independence of the matrix’s rows or columns. In this context, the determinants of 2×2 submatrices (minors) are used to capture essential relationships among the rows of the given 2×3 matrix, indicating which combinations of columns yield invertible submatrices. This idea is central in assessing and constructing inverses or pseudo?inverses in non-square matrices.

Matrix Rank and Invertibility

Matrix rank refers to the maximum number of linearly independent rows or columns in a matrix. The assertion involving the nonzero linear combination of minors implies that the 2×3 matrix has full row rank (rank 2). Full row rank is crucial for the existence of a left inverse since it guarantees that the rows are linearly independent, permitting the solution of equations of the form A·C = I, where I is the identity matrix.

Left Inverse of a Matrix

A left inverse of a matrix A is another matrix C such that the product CA equals the identity matrix. For a non-square matrix with more columns than rows, such as a 2×3 matrix, a left inverse exists if and only if the matrix has full row rank. The given condition, which involves a combination of determinants equaling 1, ensures that enough linear independence is present to construct such a matrix C, thereby allowing one to recover the original row structure via multiplication.

Transcript

00:01 Alrighty, folks.

00:02 So here we have three matrices, a1, a2, and a3, where a1 is equal to one -half of 0 -i -i -0.

00:10 A2 is one -half of 0 -1 -9 -0, and a3 is 1 -5 of i -0 -0 -9 -9.

00:18 We need to show that these three matrices satisfy these three equations down below, relating the commutators of the different matrices.

00:26 So i can just jump into this.

00:35 Into this, the commutator of a1 and a2 is going to be a1a2 minus a2a2a1.

00:40 So i have the matrices written out here.

00:44 And i'll note that scalers always commute with matrices.

00:50 So these one -haves, you know, we would have one -half times a matrix times one -half times another matrix.

00:59 So we can consolidate those one -haves instead into being one -quarters out in front of just the multiplication of the two matrices by themselves.

01:09 So one second here.

01:13 There we go.

01:14 So now just got to go through and do our matrix multiplication, knowing that the row i column j element of the product is going to be the dot product of the row i of the left -hand matrix with the column j of the right -hand matrix.

01:29 That's going to be here.

01:31 1 -1 is 0 times 0 plus i times 1.

01:33 So that's going to be i.

01:36 2 -1 is going to be 1 .2 .1 is going to be i times 0 or 0 times 1 that's going to be 0 .1 2 is going to be 0 times negative 1 plus i times 0 it's going to be 0 and 2 2 is going to be i times negative 1 plus 0 times 0 so that's negative i right there then we want to subtract off 1 quarter of and let's go through and do this element 1 -1 is going to be 0 times 0 plus negative 1 times i so that's going to be negative i element 2 .1 is going to be 1 times 0 plus 0 times i so that's 0.

02:14 Element 1 2 is going to be 0 times i plus negative 1 times 0.

02:19 So that's 0.

02:20 And element 2 2 is going to be 1 times i plus 0 times 0.

02:24 So that's going to be i.

02:27 So going through and doing our matrix addition or subtraction here.

02:33 So we'd have 1 quarter of i minus 1 quarter of negative i.

02:38 So that's going to be, we have out front here, what we can do is we can actually factor out that one quarter from our two matrices like that and do our matrix addition or subtraction here.

03:00 So inside we'd have i, sorry, in our first, in our 1 -1 element, we'd have i minus negative i.

03:07 So that's i plus i.

03:09 That's going to be 2i there.

03:10 Then 0 minus 0, 0 minus 0, negative i minus i...

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