Skip to main content
Contents Index Dark Mode Prev Up Next \(\require{cancel}
\newcommand{\bigcdot}{\mathbin{\large\boldsymbol{\cdot}}}
\newcommand{\basisfont}[1]{\mathcal{#1}}
\newcommand{\iddots}{{\mkern3mu\raise1mu{.}\mkern3mu\raise6mu{.}\mkern3mu \raise12mu{.}}}
\DeclareMathOperator{\RREF}{RREF}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\matrixring}{M}
\DeclareMathOperator{\poly}{P}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\nullity}{nullity}
\DeclareMathOperator{\nullsp}{null}
\DeclareMathOperator{\uppermatring}{U}
\DeclareMathOperator{\trace}{trace}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\negop}{neg}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\im}{im}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\ci}{\mathrm{i}}
\newcommand{\cconj}[1]{\bar{#1}}
\newcommand{\lcconj}[1]{\overline{#1}}
\newcommand{\cmodulus}[1]{\left\lvert #1 \right\rvert}
\newcommand{\bbrac}[1]{\bigl(#1\bigr)}
\newcommand{\Bbrac}[1]{\Bigl(#1\Bigr)}
\newcommand{\irst}[1][1]{{#1}^{\mathrm{st}}}
\newcommand{\ond}[1][2]{{#1}^{\mathrm{nd}}}
\newcommand{\ird}[1][3]{{#1}^{\mathrm{rd}}}
\newcommand{\nth}[1][n]{{#1}^{\mathrm{th}}}
\newcommand{\leftrightlinesubstitute}{\scriptstyle \overline{\phantom{xxx}}}
\newcommand{\inv}[2][1]{{#2}^{-{#1}}}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\degree}[1]{{#1}^\circ}
\newcommand{\blank}{-}
\newenvironment{sysofeqns}[1]
{\left\{\begin{array}{#1}}
{\end{array}\right.}
\newcommand{\iso}{\simeq}
\newcommand{\abray}[1]{\overrightarrow{#1}}
\newcommand{\abctriangle}[1]{\triangle #1}
\newenvironment{abmatrix}[1]
{\left[\begin{array}{#1}}
{\end{array}\right]}
\newenvironment{avmatrix}[1]
{\left\lvert\begin{array}{#1}}
{\end{array}\right\rvert}
\newcommand{\mtrxvbar}{\mathord{|}}
\newcommand{\utrans}[1]{{#1}^{\mathrm{T}}}
\newcommand{\rowredarrow}{\xrightarrow[\text{reduce}]{\text{row}}}
\newcommand{\bidentmattwo}{\begin{bmatrix} 1 \amp 0 \\ 0 \amp 1 \end{bmatrix}}
\newcommand{\bidentmatthree}{\begin{bmatrix} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \end{bmatrix}}
\newcommand{\bidentmatfour}{\begin{bmatrix} 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0
\\ 0 \amp 0 \amp 0 \amp 1 \end{bmatrix}}
\newcommand{\uvec}[1]{\mathbf{#1}}
\newcommand{\zerovec}{\uvec{0}}
\newcommand{\bvec}[2]{#1\,\uvec{#2}}
\newcommand{\ivec}[1]{\bvec{#1}{i}}
\newcommand{\jvec}[1]{\bvec{#1}{j}}
\newcommand{\kvec}[1]{\bvec{#1}{k}}
\newcommand{\injkvec}[3]{\ivec{#1} - \jvec{#2} + \kvec{#3}}
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\unorm}[1]{\norm{\uvec{#1}}}
\newcommand{\dotprod}[2]{#1 \bigcdot #2}
\newcommand{\udotprod}[2]{\dotprod{\uvec{#1}}{\uvec{#2}}}
\newcommand{\crossprod}[2]{#1 \times #2}
\newcommand{\ucrossprod}[2]{\crossprod{\uvec{#1}}{\uvec{#2}}}
\newcommand{\uproj}[2]{\proj_{\uvec{#2}} \uvec{#1}}
\newcommand{\adjoint}[1]{{#1}^\ast}
\newcommand{\matrixOfplain}[2]{{\left[#1\right]}_{#2}}
\newcommand{\rmatrixOfplain}[2]{{\left(#1\right)}_{#2}}
\newcommand{\rmatrixOf}[2]{\rmatrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\matrixOf}[2]{\matrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\invmatrixOfplain}[2]{\inv{\left[#1\right]}_{#2}}
\newcommand{\invrmatrixOfplain}[2]{\inv{\left(#1\right)}_{#2}}
\newcommand{\invmatrixOf}[2]{\invmatrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\invrmatrixOf}[2]{\invrmatrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\stdmatrixOf}[1]{\left[#1\right]}
\newcommand{\ucobmtrx}[2]{P_{\basisfont{#1} \to \basisfont{#2}}}
\newcommand{\uinvcobmtrx}[2]{\inv{P}_{\basisfont{#1} \to \basisfont{#2}}}
\newcommand{\uadjcobmtrx}[2]{\adjoint{P}_{\basisfont{#1} \to \basisfont{#2}}}
\newcommand{\coordmapplain}[1]{C_{#1}}
\newcommand{\coordmap}[1]{\coordmapplain{\basisfont{#1}}}
\newcommand{\invcoordmapplain}[1]{\inv{C}_{#1}}
\newcommand{\invcoordmap}[1]{\invcoordmapplain{\basisfont{#1}}}
\newcommand{\similar}{\sim}
\newcommand{\inprod}[2]{\left\langle\, #1,\, #2 \,\right\rangle}
\newcommand{\uvecinprod}[2]{\inprod{\uvec{#1}}{\uvec{#2}}}
\newcommand{\orthogcmp}[1]{{#1}^{\perp}}
\newcommand{\vecdual}[1]{{#1}^\ast}
\newcommand{\vecddual}[1]{{#1}^{\ast\ast}}
\newcommand{\dd}[2]{\frac{d{#1}}{d#2}}
\newcommand{\ddx}[1][x]{\dd{}{#1}}
\newcommand{\ddt}[1][t]{\dd{}{#1}}
\newcommand{\dydx}{\dd{y}{x}}
\newcommand{\dxdt}{\dd{x}{t}}
\newcommand{\dydt}{\dd{y}{t}}
\newcommand{\intspace}{\;}
\newcommand{\integral}[4]{\int^{#2}_{#1} #3 \intspace d{#4}}
\newcommand{\funcdef}[3]{#1\colon #2\to #3}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Reflections 8.7 Reflect on your understanding
The reflections activities in this section concern only square matrices.
In addition to the reflection activities below, re-read
Section 8.2 Terminology and notation . Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.
1. \(1 \times 1\) determinants.
Explain the conceptual reason we have defined the determinant of the general \(1 \times 1\) matrix \(\begin{bmatrix} a \end{bmatrix}\) to be the value of the single entry \(a\text{.}\)
2. Inductive definition of the determinant.
Describe how the definition of the determinant for an \(n \times n\) matrix depends on the determinant for \((n - 1) \times (n - 1)\) matrices (assuming \(n \ge 2\) ).
3. Minors versus cofactors.
Describe the relationship between the \(\nth[(i,j)]\) minor and the \(\nth[(i,j)]\) cofactor of a matrix for a specific pair of indices \((i,j)\text{.}\)
4. Minors/cofactors versus entries.
(a) State which entries in an \(n \times n\) matrix are involved in the computation of the \(\nth[(i,j)]\) minor of that matrix. Repeat for the computation of the \(\nth[(i,j)]\) cofactor.
(b) In particular, do the values of the \(\nth[(i,j)]\) minor and cofactor of a matrix depend on the value of the \(\nth[(i,j)]\) entry of that matrix?
5. Cofactor signs.
Describe in words the general pattern of cofactor signs in an \(n \times n\) matrix.
6. Determinants of diagonal/triangular matrices.
(a) State the simple pattern for computing the determinant of a diagonal or upper/lower triangular matrix.
(b) What conclusion(s) can be made about a diagonal or triangular matrix whose determinant is known to be zero?
