Determinants

Front 1

Which row is the determinant a linear function of

Back 1

The determinant is a linear fuction of the first row (when other rows stay fixed)

Front 2

When does the determinant change sign?

Back 2

The determinant changes sign when two rows are interchanged.

As a consequence of properties (1) and (2) the determinant is a linear function of any row (when the other rows stay fixed)

Front 3

What is the determinant of the identity matrix?

Back 3

The determinant of the identity matrix is 1.

For the 2X2 case, compute det(I). For the n*n case use induction and the definition, expanding along the first row: det(In+1) = 1*det(In) = 1*1 = 1

Front 4

If two rows are equal, what does the determinant equal?

Back 4

If two rows are equal, then the determinant is 0:

Let A be a matrix with two rows the same. Swapping these two rows doesn't change A but by property (2) it does change the determinant: det(A) = -det(A0. This forces det(A) = 0.

Front 5

What effect does adding a multiple of one row to another row have on the determinant

Back 5

Adding a multiple of one row to another row leaves the determinant unchanged:

Express the matrix A in terms of its rows, and add a multiple of the first row into the second to get the matrix B. Use the linearity in the second row (Property 2) together with Property 4. Show this.

Front 6

What effect does a row of 0's have on det(A)

Back 6

The determinant of a matrix with a row of 0's is 0.

The determinant is linear in any row (Property 2), in particular in the row with all 0's. Any linear transformation must map 0 to 0.

Front 7

det(A) is equal to what on a triangular matrix?

Back 7

The determinant of a triangular matrix is the product of its entries along the diagonal. Show this.

Front 8

Can a zero det(A) matrix be invertible?

Back 8

A matrix is invertible if and only if its determinant is non-zero.

Let R be the reduced row echelon form of the matrix A. By Properties 1,2, and 5, we know that there is a non-zero number k with the property that det(A) = k*det(R). If A is invertible, then R = I so det(A) = k != 0. If A is not invertible then R has at least one rows of zeros, so det(A) = k * det(R) = k * 0 = 0 by Property 4.

Front 9

What is the determinant of a product equal to?

Back 9

The determinant of a product is the product of the determinants.

If E is an elementary matrix, Properties 1, 2, and 4 imply that det(EB) = det(E)det(B) for any matrix B. If A is the product of elementary matrices, then by induction on the number of matrices in this product this can be proved. Show this. If A is not the product of elementary matrices then A is not invertible, and neither is AB. Therefore, det(AB) = 0 = 0 * det(B) = det(A)det(B)

This implies that det(AB) = det(BA) even though AB might not be the same as BA.

Front 10

What does a matrix and its transpose have in common to each other?

Back 10

A matrix and its transpose have the same determinant.

If E is an elementary matrix then so is E^T, and it's easy to see that det(E) = det(E^T). Because of this fact, everything we have said about rows holds for columns.