Determinants and Row Operations

Row Operations and Determinants

Row Operations and Determinant Changes

If a row operation is performed on a matrix, the determinant changes as follows:

1. Type One Row Operation (Row Replacement): Replacing a row with itself plus a scalar multiple of another row leaves the determinant unchanged.

2. Type Two Row Operation (Row Swap): Swapping two rows multiplies the determinant by -1.

3. Type Three Row Operation (Row Scaling): Scaling a row by a factor $k$ multiplies the determinant by the same factor $k$. Scaling all $n$ rows by $k$ (i.e., multiplying the entire matrix by $k$) scales the determinant by $k^n$.

Example

Consider matrices A, B1, B2, and B3. Matrix B1 is obtained from A by replacing the first row with the first row plus two times the second row. Since this is a type one row operation, the determinant remains unchanged.

$det(A) = -2$ and $det(B1) = -2$

Matrix B2 is obtained from A by swapping the two rows. This multiplies the determinant by -1.

Matrix B3 is obtained from A by scaling the second row by a factor of two. This multiplies the determinant by two:

$det(B3) = 2 \cdot det(A) = 2 \cdot (-2) = -4$

Using Row Operations to Calculate Determinants

Row operations can be used to simplify a matrix before calculating its determinant. By keeping track of the row operations performed, the determinant of the original matrix can be recovered.

For example, given a matrix, row operations can be performed to eliminate entries and introduce zeros. A type one row operation (adding a scalar multiple of one row to another) leaves the determinant unchanged. After performing row operations, cofactor expansion can be used to calculate the determinant.

Example of Calculating Determinant Combining Row Operations and Cofactor Expansion

• Initial Cofactor Expansion: Expand along the second row. The entry in the second row, second column is 5 which is multiplied by its cofactor with a positive sign because 2+2=4 is even ($(-1)^4 = 1$). 0 entries in other columns make those terms zero.
• Row Reduction: Subtract two times the first row from the second row. Because is a type one row operation, the determinant remains unchanged.
• Further Row Reduction: Subtract two times row one from row three. Again, this type one row operation leaves the determinant completely unchanged.
• Row Interchange: Interchange row two and row three, picking up a minus sign.
• Final Calculation: Because the matrix is now upper triangular, the determinant is the product of the diagonal entries multiplied by any scalar factors out the front, which leads to the answer.

Scaling a Multiple of a Row

If a row has a common factor, you can pull the scalar factor out the front, so long as you don't change any of the other rows. For example, if we scale the first row by a factor of $\frac{1}{2}$, the resulting determinant is half of what we actually need, so we need to multiply by 2.

Type one row operations performed to eliminate entries, such as:

$R<em>2 \rightarrow R</em>2 - 5R<em>1$$R</em>3 \rightarrow R<em>3 - 7R</em>1$

leave the determinant unchanged.

Scaling a row by a factor of $-\frac{1}{4}$ requires multiplying by -4 to preserve equality.

Determinant of Triangular Matrices

For upper or lower triangular matrices, the determinant is the product of the diagonal entries. If a matrix is brought to row echelon form using row interchanges but not row scaling, the determinant of the original matrix is the product of the diagonal entries multiplied by $(-1)^k$, where $k$ is the number of row interchanges.

Invertibility and Determinants

A square matrix is invertible if and only if its determinant is non-zero. From the invertible matrix theorem, a square matrix in row echelon form is invertible if there is a pivot in every row/column. This occurs if and only if there are no zero rows, which means every diagonal entry is non-zero, leading to a non-zero determinant.

Properties of Determinants

1. Transpose: The determinant of the transpose of a matrix is equal to the determinant of the original matrix: $det(A^T) = det(A)$.
2. Multiplication: The determinant of the product of two matrices is the product of their determinants: $det(AB) = det(A) \cdot det(B)$.
3. Addition: In general, $det(A + B) \neq det(A) + det(B)$.

Cramer's Rule

Cramer's rule provides an explicit formula for the solution to $Ax = b$. The $i^{th}$ entry of the solution vector is given by

$x<em>i = \frac{det(A</em>i)}{det(A)}$

where $A_i$ is the matrix obtained by replacing the $i^{th}$ column of $A$ with $b$. Note that $A$ must be invertible for this to hold.

Cramer's rule can also be used to find an explicit formula for the inverse of a matrix A. The $(i, j)^{th}$ entry of $A^{-1}$ is given by:

$(A^{-1})<em>{ij} = \frac{C</em>{ji}}{det(A)}$

where $C_{ji}$ is the $(j, i)^{th}$ cofactor of $A$.

Geometric Interpretation of Determinants

The determinant can be interpreted as a scaling factor for areas or volumes.

• In two dimensions, the determinant of a matrix formed by two vectors is the signed area of the parallelogram spanned by those vectors. The sign indicates whether the transformation flips the plane over.
• In three dimensions, the determinant of a matrix formed by three vectors is the signed volume of the parallelepiped spanned by those vectors.
• More generally, if a linear transformation is determined by a matrix A, the area (or volume) of any region S is scaled by a factor of $|det(A)|$ after the transformation.