How unit 5 is examined
This unit covers rank, solving linear systems, eigenvalues, diagonalization and Cayley-Hamilton; no topic was asked in the 2022-2025 papers, so all are short.
Rank of a Matrix
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Definition. <mark>The rank of a matrix is the order of its largest non-zero minor, equivalently the number of non-zero rows in its row echelon form.</mark>
Key points.
- Reduce the matrix to echelon form by elementary row operations; these do not change the rank.
- For an $m \times n$ matrix, $\rho(A) \le \min(m,n)$.
- Example: $\begin{pmatrix}1&2\\2&4\end{pmatrix}$ has $R_2 \to R_2-2R_1 = 0$, so rank 1.
- A square matrix of order $n$ has rank $n$ exactly when $|A| \ne 0$.
Solution of Simultaneous Linear Equations by Elementary Transformation
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Definition. ==For $AX=B$, form the augmented matrix $[A\,|\,B]$ and reduce it by elementary row operations to echelon form, then back-substitute.==
Key points.
- Only row operations are used, so the solution set does not change.
- Each non-zero row of the echelon form gives one equation, solved from the last row upward.
- Example: $x+y=3,\ x-y=1$ gives $[A|B]=\begin{pmatrix}1&1&3\\1&-1&1\end{pmatrix}$, $R_2\to R_2-R_1$ gives $-2y=-2$, so $y=1$, $x=2$.
- A zero row with non-zero right side signals no solution.
Consistency of Equation
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Definition. ==A system $AX=B$ is consistent if it has at least one solution; this holds exactly when $\rho(A)=\rho([A|B])$.==
Key points.
- If $\rho(A)=\rho([A|B])=n$ (number of unknowns), the solution is unique.
- If $\rho(A)=\rho([A|B])<n$, there are infinitely many solutions with $n-\rho$ free variables.
- If $\rho(A)\ne\rho([A|B])$, the system is inconsistent and has no solution.
- A homogeneous system $AX=0$ is always consistent, since $X=0$ works.
Eigen Values and Eigen Vectors
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Definition. ==If $AX=\lambda X$ for a non-zero vector $X$, then $\lambda$ is an eigenvalue and $X$ its eigenvector of $A$.==
Key points.
- Eigenvalues are the roots of the characteristic equation $|A-\lambda I|=0$.
- For each $\lambda$, solve $(A-\lambda I)X=0$ to get the eigenvector.
- Sum of eigenvalues equals the trace, and their product equals $|A|$.
- Example: $A=\begin{pmatrix}2&1\\1&2\end{pmatrix}$ gives $(2-\lambda)^2-1=0$, so $\lambda=1,3$ with vectors $(1,-1)^T$ and $(1,1)^T$.
Diagonalization of Matrices
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Definition. ==A square matrix $A$ is diagonalizable if there is an invertible $P$ with $P^{-1}AP=D$, a diagonal matrix.==
Key points.
- The columns of $P$ are the eigenvectors of $A$ and the diagonal of $D$ holds the eigenvalues in the same order.
- $A$ is diagonalizable exactly when it has $n$ linearly independent eigenvectors, which is guaranteed if all eigenvalues are distinct.
- Powers become easy: $A^k=PD^kP^{-1}$.
- Example: for the matrix above, $P=\begin{pmatrix}1&1\\-1&1\end{pmatrix}$ and $D=\mathrm{diag}(1,3)$.
Cayley-Hamilton theorem and its applications to find inverse
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Definition. <mark>Every square matrix satisfies its own characteristic equation.</mark>
Key points.
- If $|A-\lambda I|=\lambda^n+a_1\lambda^{n-1}+\dots+a_n$, then $A^n+a_1A^{n-1}+\dots+a_nI=0$.
- To find the inverse, multiply the equation by $A^{-1}$ and solve for $A^{-1}$ (needs $a_n\ne0$).
- Example: $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ has $\lambda^2-5\lambda-2=0$, so $A^2-5A-2I=0$ and $A^{-1}=\tfrac12(A-5I)=\begin{pmatrix}-2&1\\1.5&-0.5\end{pmatrix}$.
- It also reduces higher powers of $A$ to lower ones.
Last-minute revision
- Rank = order of largest non-zero minor = number of non-zero rows in echelon form.
- Row operations preserve rank and the solution set.
- Consistent iff $\rho(A)=\rho([A|B])$.
- Equal to $n$ means unique solution; less than $n$ means infinitely many.
- Eigenvalues solve $|A-\lambda I|=0$.
- Sum of eigenvalues is the trace; product is the determinant.
- $P^{-1}AP=D$ with eigenvectors as columns of $P$.
- $A^k=PD^kP^{-1}$.
- Cayley-Hamilton: $A$ satisfies its characteristic equation.
- $A^{-1}$ from Cayley-Hamilton: multiply the equation by $A^{-1}$.
Memory hooks
- Rank test: "same rank, some solution; same as $n$, only one".
- Trace adds, determinant multiplies (eigenvalues).
- $PDP^{-1}$: eigenvectors in, eigenvalues on the diagonal.
- Cayley-Hamilton: replace $\lambda$ by $A$, the constant by $I$.
Coverage checklist
- Rank of a Matrix: definition, echelon method (no past questions).
- Solution of Simultaneous Linear Equations by Elementary Transformation: augmented matrix (no past questions).
- Consistency of Equation: rank condition (no past questions).
- Eigen Values and Eigen Vectors: characteristic equation (no past questions).
- Diagonalization of Matrices: $P^{-1}AP=D$ (no past questions).
- Cayley-Hamilton theorem and its applications to find inverse: theorem, inverse (no past questions).