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10 Key properties of Determinant

• Rule 1: The determinant of the n by n identity matrix is 1. \(\left|\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right|=1 \quad \text { and } \quad\left|\begin{array}{lll} 1 & & \\ & \ddots & \\ & & 1 \end{array}\right|=1 \text {. }\)

• Rule 2: The determinant changes sign when two rows are exchanged (sign reversal): \(\text { Check: }\left|\begin{array}{ll} c & d \\ a & b \end{array}\right|=-\left|\begin{array}{ll} a & b \\ c & d \end{array}\right| \quad \text { (both sides equal } b c-a d \text { ). }\) Because of this rule, we can find det P for any permutation matrix. Just exchange rows of I until you reach P. Then det P = +1 for an even number of row exchanges and det P = -l for an odd number. The third rule has to make the big jump to the determinants of all matrices.

• Rule 3: The determinant is a linear function of each row separately (all other rows stay fixed). \(\begin{aligned} & \left|\begin{array}{cc} t a & t b \\ c & d \end{array}\right|=t\left|\begin{array}{ll} a & b \\ c & d \end{array}\right| \\ & \left|\begin{array}{cc} a+a^{\prime} & b+b^{\prime} \\ c & d \end{array}\right|=\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|+\left|\begin{array}{cc} a^{\prime} & b^{\prime} \\ c & d \end{array}\right| . \end{aligned}\) \(\left|\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right|=2^2=4 \quad \text { and } \quad\left|\begin{array}{ll} t & 0 \\ 0 & t \end{array}\right|=t^2 \text {. }\)

• Rule 4: If two rows of A are equal, then $let A = 0$.

Rule 4 follows from rule 2.

• Rule 5: Subtracting a multiple of one row from another row leaves det A unchanged. \(\begin{aligned} & \ell \text { times row } 1 \\ & \text { from row } 2 \end{aligned} \quad\left|\begin{array}{cc} a & b \\ c-\ell a & d-\ell b \end{array}\right|=\left|\begin{array}{ll} a & b \\ c & d \end{array}\right| .\)
  from Rule 3 (linearity).

Conclusion: The determinant is not changed by the usual elimination steps from A to U. Thus det A equals det U. If we can find determinants of triangular matrices U, we can find determinants of all matrices A. Every row exchange reverses the sign, so always det A= ± det U. Rule 5 has narrowed the problem to triangular matrices.

• Rule 6: A matrix with a row of zeros has det A = 0. \(\text { Row of zeros } \quad\left|\begin{array}{ll} 0 & 0 \\ c & d \end{array}\right|=0 \quad \text { and } \quad\left|\begin{array}{ll} a & b \\ 0 & 0 \end{array}\right|=0 \text {. }\)

• Rule 7: If A is triangular then $\operatorname{det} A=a_{11} a_{22} \cdots a_{n n}=$ product of diagonal entries. \(\text { Triangular } \quad\left|\begin{array}{ll} a & b \\ 0 & d \end{array}\right|=a d \quad \text { and also }\left|\begin{array}{ll} a & 0 \\ c & d \end{array}\right|=a d \text {. }\) further : \(\text { Diagonal matrix } \operatorname{det}\left[\begin{array}{cccc} a_{11} & & & 0 \\ & a_{22} & & \\ & & \ddots & \\ 0 & & & a_{n n} \end{array}\right]=\left(a_{11}\right)\left(a_{22}\right) \cdots\left(a_{n n}\right) \text {. }\)

• Rule 8: If $A$ is singular then $\operatorname{det} A=0$. If $A$ is invertible then $\operatorname{det} A \neq 0$. \(\text { Singular } \quad\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \text { is singular if and only if } a d-b c=0 \text {. }\) \(\text { Multiply pivots } \quad \operatorname{det} A= \pm \operatorname{det} U= \pm \text { (product of the pivots). }\) \(\text { The determinant is }\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|=\left|\begin{array}{cc} a & b \\ 0 & d-(c / a) b \end{array}\right|=a d-b c \text {. }\) \(\text { If } P A=L U \text { then } \operatorname{det} P \operatorname{det} A=\operatorname{det} L \operatorname{det} U \text { and } \operatorname{det} A= \pm \operatorname{det} U \text {. }\)

• Rule 9: The determinant of $A B$ is $\operatorname{det} A$ times $\operatorname{det} B:|A B|=|A||B|$. \(\text { Product rule } \quad\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|\left|\begin{array}{ll} p & q \\ r & s \end{array}\right|=\left|\begin{array}{ll} a p+b r & a q+b s \\ c p+d r & c q+d s \end{array}\right| \text {. }\) \(A \text { times } A^{-1} \quad A A^{-1}=I \text { so } \quad(\operatorname{det} A)\left(\operatorname{det} A^{-1}\right)=\operatorname{det} I=1 .\)

• Rule 10: The transpose $A^{\mathrm{T}}$ has the same determinant as $A$. \(\text { Transpose }\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|=\left|\begin{array}{ll} a & c \\ b & d \end{array}\right| \text { since both sides equal } a d-b c \text {. }\)

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Important comment on columns