2.2 Matrices

Learning objectives

2.2.1Matrix equations• formulate a problem involving the solution of 3 linear simultaneous equations in 3 unknowns as a problem involving the solution of a matrix equation, or vice versa 2.2.2Determinants and inverses• understand the cases that may arise concerning the consistency or inconsistency of 3 linear simultaneous equations, relate them to the singularity or otherwise of the corresponding matrix, solve consistent systems, and interpret geometrically in terms of lines and planes e.g. three planes meeting in a common point, or in a common line, or having no common points.2.2.3Eigen terminology• understand the terms 'characteristic equation', 'eigenvalue' and 'eigenvector', as applied to square matrices Including use of the definition Ae = λe to prove simple properties, e.g. that λn is an eigenvalue of An.2.2.4Eigenvalues and eigenvectors• find eigenvalues and eigenvectors of 2 x 2 and 3 x 3 matrices Restricted to cases where the eigenvalues are real and distinct.2.2.5Eigenvalues and eigenvectors• express a square matrix in the form QDQ-1, where D is a diagonal matrix of eigenvalues and Q is a matrix whose columns are eigenvectors, and use this expression e.g. in calculating powers of 2 x 2 or 3 x 3 matrices.2.2.6Eigen terminology• use the fact that a square matrix satisfies its own characteristic equation. e.g. in finding successive powers of a matrix or finding an inverse matrix; restricted to 2 x 2 or 3 x 3 matrices only.

$The system$ a_{11}x+a_{12}y+a_{13}z=b_1,\quad a_{21}x+a_{22}y+a_{23}z=b_2,\quad a_{31}x+a_{32}y+a_{33}z=b_3 $is exactly$ A\mathbf x=\mathbf b,\qquad A=(a_{ij}),\quad\mathbf x=\begin{pmatrix}x\y\z\end{pmatrix},\quad\mathbf b=\begin{pmatrix}b_1\b_2\b_3\end{pmatrix}.

\begin{aligned}2x-y+z&=4\x+3y-2z&=5\3x+y+z&=6\end{aligned}\quad\Longleftrightarrow\quad \begin{pmatrix}2&-1&1\1&3&-2\3&1&1\end{pmatrix}\begin{pmatrix}x\y\z\end{pmatrix}=\begin{pmatrix}4\5\6\end{pmatrix}.

To reverse the process, multiply each row of A by the unknown column and equate it to the corresponding entry of b. If the problem supplies quantities rather than x,y,z, define the three unknowns first and keep that column order unchanged throughout.

The constants form b, not an extra coefficient column inside A. A row represents one equation; a column represents one unknown across all equations.

For $A\mathbf x=\mathbf b$ with three unknowns:

Algebraic evidence	Solutions	Plane geometry
$\det A\ne0$	One, $\mathbf x=A^{-1}\mathbf b$	Three planes meet at one point
$\det A=0$ and $\operatorname{rank}A=\operatorname{rank}[A\|\mathbf b]<3$	Infinitely many	Common line when rank 2; a common plane/more freedom when rank is lower
$\operatorname{rank}A<\operatorname{rank}[A\|\mathbf b]$	None (inconsistent row such as $0=1$ )	The three planes have no common point

First use $\det A$ to test uniqueness. If it is zero, row-reduce the augmented matrix $[A|\mathbf b]$ . A zero row on both sides removes a constraint and leaves free parameter(s); a zero coefficient row with a non-zero right side is a contradiction. Solve every consistent case and state the geometric interpretation.

$The equations$ x+y+z=1,\quad2x+2y+2z=2,\quad x-y+z=0 $have one dependent row and two independent constraints, so they are consistent with one free parameter: three planes share a line. Replacing the second right side by 3 creates $0=1$ after elimination, so there is no common point.$

Singular means ‘not uniquely solvable’, not automatically ‘no solution’. Determinant alone cannot distinguish a common line from inconsistency; the augmented system must be examined.

A non-zero vector e is an eigenvector of A with eigenvalue $\lambda$ when $A\mathbf e=\lambda\mathbf e$ . Rearranging gives $(A-\lambda I)\mathbf e=0$ . A non-zero solution exists exactly when $A-\lambda I$ is singular, so $\det(A-\lambda I)=0$ is the characteristic equation.

All non-zero scalar multiples of e represent the same eigendirection. A positive eigenvalue preserves direction, a negative one reverses it, and eigenvalue zero collapses that direction; the zero vector itself is excluded because it would satisfy every lambda.

$If $A\mathbf e=\lambda\mathbf e$, then$ A^2\mathbf e=A(\lambda\mathbf e)=\lambda A\mathbf e=\lambda^2\mathbf e. $Repeating (or inducting) gives$ A^n\mathbf e=\lambda^n\mathbf e, $so e is an eigenvector of $A^n$ with eigenvalue $\lambda^n$.$

The characteristic equation finds eigenvalues, not eigenvectors. Each root lambda must still be substituted into $(A-\lambda I)\mathbf e=0$ to obtain a non-zero eigendirection.

For a 2x2 or 3x3 matrix A: form $\det(A-\lambda I)=0$ without row-operating A first; solve for the real distinct eigenvalues; for each lambda solve $(A-\lambda I)\mathbf e=0$ ; choose any convenient non-zero scalar representative; finally verify $A\mathbf e=\lambda\mathbf e$ .

$For$ A=\begin{pmatrix}1&1&0\0&2&1\0&0&3\end{pmatrix},\det(A-\lambda I)=(1-\lambda)(2-\lambda)(3-\lambda), $so the real distinct eigenvalues are 1, 2 and 3.$

lambda	Solve $(A-\lambda I)e=0$	One eigenvector
1	$y=z=0$	$(1,0,0)^T$
2	$y=x, z=0$	$(1,1,0)^T$
3	$y=2x, z=2x$	$(1,2,2)^T$

This syllabus restricts computation here to real distinct eigenvalues. Diagonal entries reveal eigenvalues immediately only for triangular matrices; the characteristic determinant remains the general method.

$If $A\mathbf q_i=\lambda_i\mathbf q_i$ and the eigenvectors form a basis, set$ Q=(\mathbf q_1\ \mathbf q_2\ \cdots),\qquad D=\operatorname{diag}(\lambda_1,\lambda_2,\ldots). $Then $AQ=QD$, so$ A=QDQ^{-1}.

$Adjacent factors cancel:$ (QDQ^{-1})^n=QD^nQ^{-1}, $and $D^n$ is obtained by raising each diagonal eigenvalue to n. The order of columns in Q must match the order of entries in D.$

$For the previous matrix,$ Q=\begin{pmatrix}1&1&1\0&1&2\0&0&2\end{pmatrix},\quad D=\operatorname{diag}(1,2,3),\quad Q^{-1}=\begin{pmatrix}1&-1&1/2\0&1&-1\0&0&1/2\end{pmatrix}. $Hence$ A^n=\begin{pmatrix}1&2^n-1&(3^n-2\cdot2^n+1)/2\0&2^n&3^n-2^n\0&0&3^n\end{pmatrix}.

Check $n=0$ gives I and $n=1$ gives A. Do not power Q and $Q^{-1}$ separately; their cancellation is what makes only D receive the exponent.

$Cayley-Hamilton states: if$ p(\lambda)=\lambda^m+c_{m-1}\lambda^{m-1}+\cdots+c_1\lambda+c_0 $is the characteristic polynomial of A, then$ p(A)=A^m+c_{m-1}A^{m-1}+\cdots+c_1A+c_0I=0.

Rearrange the identity to replace the highest power, then repeat to reduce successive powers to degree below m. If A is non-singular, multiply by a suitable negative power or isolate the constant-I term to express $A^{-1}$ as a polynomial in A.

$For$ A=\begin{pmatrix}2&1\1&1\end{pmatrix},p(\lambda)=\lambda^2-3\lambda+1, $so$ A^2-3A+I=0. $Therefore$ A^3=3A^2-A=8A-3I, $and multiplying the identity by $A^{-1}$ gives$ A^{-1}=3I-A=\begin{pmatrix}1&-1\-1&2\end{pmatrix}.

Replace scalar 1 by I and scalar powers by matrix powers; order is harmless only because every term is a polynomial in the same A. This objective is Cayley-Hamilton—not merely the trace/product eigenvalue check.

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—2.3 DifferentiationRead

2.2 Matrices

Learning objectives

Rows are equations and columns are variables

A zero determinant starts the consistency test—it does not finish it

The characteristic equation finds directions scaled by a matrix

Find every eigenpair by solving one singular system per root

Match eigenvector columns to diagonal entries before taking powers

Substitute the matrix into its own characteristic polynomial

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