Table of Contents

## How does swapping rows affect the determinant?

Swapping those rows doesn’t change the determinant, but at the same time does change its sign. The only number unchanged by changing its sign is 0, so the determinant must be 0. The value of a determinant with two equal rows must be 0.

**Do row and column operations change determinant?**

The answer: yes, if you’re careful. Row operations change the value of the determinant, but in predictable ways. If you keep track of those changes, you can use row operations to evaluate determinants.

**Does matrix determinant change with row operations?**

Computing a Determinant Using Row Operations If two rows of a matrix are equal, the determinant is zero. If two rows of a matrix are interchanged, the determinant changes sign. If a multiple of a row is subtracted from another row, the value of the determinant is unchanged.

### What happens to the determinant when you transpose a matrix?

The determinant of a square matrix is the same as the determinant of its transpose. The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix.

**How does swapping columns change the determinant?**

If we add a row (column) of A multiplied by a scalar k to another row (column) of A, then the determinant will not change. If we swap two rows (columns) in A, the determinant will change its sign. det(A)=det(AT).

**Does scaling a matrix change the determinant?**

The determinant is multiplied by the scaling factor.

#### How does the determinant change?

- The value of the determinant of a matrix doesn’t change if we transpose this matrix (change rows to columns)
- If we multiply a scalar to a matrix A, then the value of the determinant will change by a factor !
- This makes sense, since we are free to choose by which row or column we will expand the determinant.

**How does scaling a matrix affect the determinant?**

The effect of scaling a matrix. Since a linear transformation can always be written as T(x)=Ax for some matrix A, applying a linear transformation to a vector x is the same thing as multiplying by a matrix. In one dimension, the effect of doubling every vector would simply double the expansion of length by ˜T.

**What happens to determinant when matrix is inverse?**

The determinant of the inverse of an invertible matrix is the inverse of the determinant: det(A-1) = 1 / det(A) [6.2. Similar matrices have the same determinant; that is, if S is invertible and of the same size as A then det(S A S-1) = det(A). [6.2.

## Does swapping columns change the determinant?

If we add a row (column) of A multiplied by a scalar k to another row (column) of A, then the determinant will not change. If we swap two rows (columns) in A, the determinant will change its sign.

**How do you find the determinant of a column matrix?**

Expanding to Find the Determinant

- Pick any row or column in the matrix. It does not matter which row or which column you use, the answer will be the same for any row.
- Multiply every element in that row or column by its cofactor and add. The result is the determinant.

**How do you change the determinant of a column?**

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant resu… Stack Exchange Network

### Does the value of the determinant of a matrix change when transposing?

The value of the determinant of a matrix doesn’t change if we transpose this matrix (change rows to columns) ·ais a scalar, Ais n´nmatrix If we multiply a scalar to a matrix A, then the value of the determinant will change by a factor !

**How do you expand a matrix with determinants?**

You can expand on any column or row. You use a checkerboard pattern to ﬁgure the signs: + − + − − + − + + − + − − + − + and the “cross out” a row and a columns, several times, to write the big determi- nant as a sum/difference of many (many) smaller determinants. Again, see the book.

**How does swapping 2 rows invert the determinant?**

Swapping 2 rows inverts the sign of the determinant. For any square matrix you can generalize the proof of swapping two rows (or columns) being equivalent to swapping the sign of the determinant by using the axiom that the determinant is invariant under elementary row (or column) operations. Consider a Matrix A ∈ F n × n as shown below;