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Which row is the determinant a linear function of

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


When does the determinant change sign?

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) 

What is the determinant of the identity matrix?

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 

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

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. 

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

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. 

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

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. 

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

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


Can a zero det(A) matrix be invertible?

A matrix is invertible if and only if its determinant is nonzero.
Let R be the reduced row echelon form of the matrix A. By Properties 1,2, and 5, we know that there is a nonzero 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. 

What is the determinant of a product equal to?

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. 

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

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. 