Determinants · Definition

Determinant

linear algebradeterminants

Before you start you need to know

The determinant sums up in a single number whether a square matrix is invertible and how much it changes volumes.

Definition

The determinant of a square matrix is a number that measures how much the linear transformation it defines scales volumes.

\[\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\]

For a system of two equations in two unknowns, a unique solution exists exactly when \(ad-bc\ne0\). That number, which comes from eliminating one unknown, is the determinant of the coefficient matrix. The idea was generalized to \(n\times n\) as a single quantity that decides whether the system \(A\mathbf{x}=\mathbf{b}\) has a unique solution for every \(\mathbf{b}\).

The geometric interpretation gives the meaning: the columns of a \(2\times2\) matrix span a parallelogram whose area is \(|\det A|\); in \(3\times3\) it is the volume of the parallelepiped. The sign tells whether the transformation preserves or reverses orientation. If the determinant is 0, the parallelogram collapses to a segment: the columns are dependent and the matrix is not invertible.

Guiding example: \(\det\begin{pmatrix}2&1\\5&3\end{pmatrix}=2\cdot3-1\cdot5=1\), so the matrix is invertible and preserves areas. Typical mistakes: subtracting in the wrong order (\(bc-ad\)), or believing \(\det(A+B)=\det A+\det B\), which is false; what does hold is \(\det(AB)=\det A\det B\). For triangular matrices it is enough to multiply the diagonal, and the next concept shows how to compute larger determinants by cofactors. For a \(2\times2\) matrix the determinant is \(ad-bc\), and for a triangular matrix it is the product of its diagonal.

Think about it

If \(\det A=0\), what can be said about the columns of \(A\)?

They are linearly dependent, which is why \(A\) is not invertible.

Is it true that \(\det(2A)=2\det A\) for a \(3\times3\) matrix?

No: each of the 3 rows is multiplied by 2, so \(\det(2A)=2^3\det A=8\det A\).

Exercises 10

  1. Compute \(\det A\) for \(A=\begin{pmatrix}3 & 1 \\ 2 & 4\end{pmatrix}\).
  2. Compute the determinant of \(\begin{pmatrix}2 & -1 \\ 5 & 3\end{pmatrix}\).
  3. Compute the determinant of the upper triangular matrix \(\begin{pmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6\end{pmatrix}\).
  4. For which value of \(k\) is \(\det\begin{pmatrix}k & 2 \\ 3 & 6\end{pmatrix}=0\)?
  5. What is the determinant of the identity matrix \(I_3\)?
  6. Compute \(\det A\) using the rule of Sarrus for \(A=\begin{pmatrix}2 & 0 & 1 \\ 1 & 3 & 2 \\ 0 & 1 & 4\end{pmatrix}\).
  7. The parallelogram spanned by the vectors \((3,1)\) and \((1,2)\) has area:
  8. Find all values of \(x\) such that \(\det\begin{pmatrix}x & 1 \\ 4 & x\end{pmatrix}=0\).
  9. Let \(A\) be a \(2\times2\) matrix with \(\det A=7\). What is \(\det(2A)\)?
  10. Compute \(\det\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}\) and decide what it says about the rows.

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §5.1
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §10.A