Table of Contents

- 1 Can we expand determinant in any row or column?
- 2 Does value of determinant change with row operations?
- 3 What is expansion in determinants?
- 4 Is value of determinant same?
- 5 What are the important properties of determinants and give each definition?
- 6 Do inverse matrices have the same determinant?
- 7 What happens when two rows of a determinant are interchanged?
- 8 How do you expand a matrix with determinants?

## Can we expand determinant in any row or column?

Expanding the n×n matrix along any row or column always gives the same answer, which is the determinant.

## Does value of 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 of a if a multiple of a row is added to another row?**

Therefore, when we add a multiple of a row to another row, the determinant of the matrix is unchanged. Note that if a matrix A contains a row which is a multiple of another row, det(A) will equal 0. By Theorem [thm:addingmultipleofrow], we can add the first row to the second row, and the determinant will be unchanged.

**How do you expand a determinant along a row?**

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.

### What is expansion in determinants?

In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1) submatrices of B.

### Is value of determinant same?

Determinant evaluated across any row or column is same. If all the elements of a row (or column) are zeros, then the value of the determinant is zero.

**Why does swapping rows change 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.

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

Proof by induction that transposing a matrix does not change its determinant.

#### What are the important properties of determinants and give each definition?

There are 10 main properties of determinants which include reflection property, all-zero property, proportionality or repetition property, switching property, scalar multiple property, sum property, invariance property, factor property, triangle property, and co-factor matrix property.

#### Do inverse matrices have the same determinant?

The determinant of the inverse of an invertible matrix is the inverse of the determinant: det(A-1) = 1 / det(A) [6.2. 6, page 265]. 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).

**Are determinants always positive?**

The determinant of a matrix is not always positive.

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

If you multiply a row or column by a non-zero constant, the determinant is multiplied by that same non-zero constant. If you multiply a row or column by a non-zero constant and add it to another row or column, replacing that row or column, there is no change in the determinant. That last operation is equivalent to pivoting on a one!

## What happens when two rows of a determinant are interchanged?

If any two rows (or columns) of a determinant are interchanged, then the sign of determinant changes. Similarly, we can verify the result by interchanging any two columns.

## 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.

**What happens when equimultiples are added to a determinant?**

If the equimultiples of corresponding elements of other rows (or columns) are added to every element of any row or column of a determinant, then the value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation R i → R i + k R j or C i → C i + k C j .