Determinants will be an incredibly useful tool in quickly determining several important properties of square matrices. We will first look at how to compute determinants and later outline the important properties that determinants have. While some of you may have been taught some rules for how to compute determinants of 2 by 2 and 3 by 3 matrices, I encourage you to understand how to compute determinants in general.
Subsection2.5.1Computing Determinants
Definition2.5.1.
The determinant is a function from \(n\) by \(n\) matrices to the real numbers (\(\det:M_{n \times n} \rightarrow \mathbb{R}\)).
If \(A\) is a 1 by 1 matrix, \(A=[A_{1,1}]\text{,}\) then \(\det(A)=A_{1,1}\text{.}\)
For \(n \geq 2\text{,}\) the determinant of a \(n\) by \(n\) matrix is given by the following formula in terms of determinants of \((n-1)\) by \((n-1)\) matrices:
where \(A^*_{i,j}\) is the \((n-1)\) by \((n-1)\) matrix obtained by deleting the \(i\)-th row and \(j\)-th column of \(A\text{.}\) The term \((-1)^{i+j} \det(A^*_{i,j})\) is called the \((i,j)\) cofactor of \(A\).
The above definition uses cofactor expansion along the first row.
Investigation2.5.1.
In this question, we will unpack the determinant formula above for a 2 by 2 matrix \(A=\begin{bmatrix} a\amp b \\c\amp d \end{bmatrix}\text{.}\)
(a)
Rather than using the summation notation of the formula above, write out the two terms in \(\det(A)\text{.}\)
(b)
\(A^*_{1,1}=\)
(c)
\(A^*_{1,2}=\)
(d)
\(A_{1,1}=\)
(e)
\(A_{1,2}=\)
(f)
\((-1)^{1+1}=\)
(g)
\((-1)^{1+2}=\)
(h)
\(\det(A)=\)
Your answer to the previous problem will be useful in calculating determinants of 3 by 3 matrices. We will use the theorem below without proving it.
Theorem2.5.2.
The determinant can be computed by cofactor expansion along any row or column. Specifically the cofactor expansion along the \(k\)-th row is given by
Use cofactor expansion along the first column of \(A=\begin{bmatrix} a\amp b\amp c\\d\amp e\amp f\\g\amp h\amp i \end{bmatrix}\) to compute \(\det(A)\text{.}\)
Checkpoint2.5.4.
Use cofactor expansion along the second row of \(A=\begin{bmatrix} a\amp b\amp c\\d\amp e\amp f\\g\amp h\amp i \end{bmatrix}\) to compute \(\det(A)\text{.}\) Did you get the same answer as the previous question?
Checkpoint2.5.5.
Compute the determinant of \(B=\begin{bmatrix}g\amp h\amp i \\d\amp e\amp f\\ a\amp b\amp c \end{bmatrix}\text{.}\) How does your answer compare with the previous problem?
Checkpoint2.5.6.
Compute the determinant of \(C=\begin{bmatrix} a\amp b\amp c\\3d\amp 3e\amp 3f\\g\amp h\amp i \end{bmatrix}\text{.}\)
Checkpoint2.5.7.
Compute the determinant of \(D=\begin{bmatrix} a+kd\amp b+ke\amp c+kf\\d\amp e\amp f\\g\amp h\amp i \end{bmatrix}\text{.}\)
Checkpoint2.5.8.
Compute the determinant of the following matrices:
Prove that if \(A\) has a row of zeros, then \(\det(A)=0\text{.}\)
Investigation2.5.4.
Prove that \(\det(Id_n)=1\text{.}\)
We will now state several useful properties of determinants. We will defer the proofs until later in the course. You may use these theorems unless a problem specifically asks you to prove one of them, in which case, the problem will note that you may not use the theorem to prove it.
Theorem2.5.9.
The determinants of elementary matrices have the following values:
If \(E_1\) multiplies a row by a scalar \(\alpha\text{,}\) then \(\det(E_1) = \alpha\text{.}\)
If \(E_2\) adds \(\alpha\) times a row to another row, then \(\det(E_2) = 1\text{.}\)
If \(E_3\) swaps two rows, then \(\det(E_3) = -1\text{.}\)
Theorem2.5.10.
If \(A\) and \(B\) are \(n\) by \(n\text{,}\) then \(\det(AB)=\det(A) \det(B)\text{.}\)
If \(A\) is \(n\) by \(n\) and \(k\) is a scalar, then \(\det(kA)=k^n \det(A)\text{.}\)
The determinant of an upper or lower triangular matrix is the product of its diagonal entries.
