GATE DA & AI Engineering Linear Algebra Syllabus

Define an orthogonal matrix $Q$.

A square matrix whose columns (and rows) are orthonormal vectors, satisfying $Q^{T}Q=QQ^{T}=I$. Equivalently $Q^{-1}=Q^{T}$.

What is the determinant of an orthogonal matrix?

$\det(Q)=\pm 1$. ($+1$ for a proper rotation, $-1$ for an improper one such as a reflection.)

State the length-preserving property of an orthogonal matrix $Q$.

$Q$ preserves norms and inner products: $\lVert Q\vec{x}\rVert = \lVert \vec{x}\rVert$ and $(Q\vec{x})^{T}(Q\vec{y})=\vec{x}^{T}\vec{y}$ for all $\vec{x},\vec{y}$.

What are the possible magnitudes of the eigenvalues of an orthogonal matrix?

All eigenvalues have absolute value $1$ ($|\lambda|=1$); they may be complex but lie on the unit circle.

If $Q_1$ and $Q_2$ are orthogonal matrices of the same size, is $Q_1Q_2$ orthogonal?

Yes. $(Q_1Q_2)^{T}(Q_1Q_2)=Q_2^{T}Q_1^{T}Q_1Q_2=Q_2^{T}Q_2=I$, so the product of orthogonal matrices is orthogonal.

Define an idempotent matrix.

A square matrix $A$ satisfying $A^{2}=A$. Applying it repeatedly gives the same result: $A^{k}=A$ for all $k\geq 1$.

What are the only possible eigenvalues of an idempotent matrix?

$0$ and $1$. From $A^{2}=A$ we get $\lambda^{2}=\lambda$, so $\lambda(\lambda-1)=0$.

For an idempotent matrix $A$, how do its trace and rank relate?

$\operatorname{rank}(A)=\operatorname{trace}(A)$, because the trace equals the sum of eigenvalues, which are $1$'s (multiplicity = rank) and $0$'s.

How is an idempotent matrix related to a projection matrix?

Every projection matrix is idempotent ($P^2=P$). An idempotent matrix represents a projection (possibly oblique); if it is additionally symmetric, it is an orthogonal projection.

If $A$ is idempotent, show that $I-A$ is also idempotent.

$(I-A)^{2}=I-2A+A^{2}=I-2A+A=I-A$. Hence $I-A$ is idempotent.

Which idempotent matrices are also invertible, and what must they equal?

Only the identity. If $A^{2}=A$ and $A^{-1}$ exists, then $A=A^{-1}A^{2}=A^{-1}A=I$. So the only invertible idempotent matrix is $I$.

What is a partitioned (block) matrix?

A matrix subdivided into smaller submatrices (blocks) by horizontal and vertical lines, e.g. $A=\begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\end{pmatrix}$, allowing operations to be carried out block-wise.

State the rule for multiplying two conformably partitioned matrices.

Treat blocks as scalar entries: $\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}E&F\\G&H\end{pmatrix}=\begin{pmatrix}AE+BG & AF+BH\\ CE+DG & CF+DH\end{pmatrix}$, provided the block dimensions are compatible for each product.

Give the determinant of a block triangular matrix $\begin{pmatrix} A & B\\ 0 & D\end{pmatrix}$ with square $A,D$.

$\det\begin{pmatrix} A & B\\ 0 & D\end{pmatrix}=\det(A)\,\det(D)$. The same holds for block lower-triangular matrices.

What is the inverse of a block diagonal matrix $\begin{pmatrix} A & 0\\ 0 & D\end{pmatrix}$ with $A,D$ invertible?

$\begin{pmatrix} A^{-1} & 0\\ 0 & D^{-1}\end{pmatrix}$. Each invertible diagonal block is simply inverted.

Define the Schur complement of $A$ in $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ (with $A$ invertible).

The Schur complement is $M/A = D - CA^{-1}B$. Then $\det(M)=\det(A)\,\det(D-CA^{-1}B)$.

What is the transpose of a partitioned matrix $\begin{pmatrix}A&B\\C&D\end{pmatrix}$?

$\begin{pmatrix}A^{T}&C^{T}\\B^{T}&D^{T}\end{pmatrix}$. Transpose each block and swap the off-diagonal blocks' positions.

Write the general form of a system of $m$ linear equations in $n$ unknowns in matrix notation.

$A\vec{x}=\vec{b}$, where $A$ is the $m\times n$ coefficient matrix, $\vec{x}\in\mathbb{R}^n$ the unknowns, and $\vec{b}\in\mathbb{R}^m$ the right-hand side.

What distinguishes a homogeneous linear system, and what solution does it always possess?

A homogeneous system has $\vec{b}=\vec{0}$, i.e. $A\vec{x}=\vec{0}$. It always has at least the trivial solution $\vec{x}=\vec{0}$, so it is always consistent.

State the Rouché–Capelli consistency condition for $A\vec{x}=\vec{b}$.

The system is consistent (has at least one solution) if and only if $\operatorname{rank}(A)=\operatorname{rank}([A\,|\,\vec{b}])$ (rank of coefficient matrix equals rank of augmented matrix).

For $A\vec{x}=\vec{b}$ with $n$ unknowns, give the rank conditions for: unique solution, infinitely many solutions, and no solution.

Let $r=\operatorname{rank}(A)$, $r'=\operatorname{rank}([A|\vec{b}])$. Unique: $r=r'=n$. Infinitely many: $r=r'<n$. No solution: $r<r'$ (i.e. $r' = r+1$).

For a homogeneous system $A\vec{x}=\vec{0}$ with $n$ unknowns, when does it have non-trivial solutions?

When $\operatorname{rank}(A)=r<n$. There are then $n-r$ free variables and infinitely many solutions. (For a square $A$: non-trivial solutions exist $\iff \det(A)=0$.)

How many free variables does a consistent system $A\vec{x}=\vec{b}$ have, and what does this equal?

Number of free variables $=n-r$, where $n$ is the number of unknowns and $r=\operatorname{rank}(A)$. This is the dimension of the null space (nullity).

State the rank–nullity theorem for an $m\times n$ matrix $A$.

$\operatorname{rank}(A)+\operatorname{nullity}(A)=n$, where $n$ is the number of columns and $\operatorname{nullity}(A)=\dim N(A)$.

Describe the structure of the general solution of a consistent non-homogeneous system $A\vec{x}=\vec{b}$.

$\vec{x}=\vec{x}_p+\vec{x}_h$, the sum of one particular solution $\vec{x}_p$ of $A\vec{x}=\vec{b}$ and any solution $\vec{x}_h$ of the homogeneous system $A\vec{x}=\vec{0}$.

What is the goal and basic idea of Gaussian elimination?

To solve $A\vec{x}=\vec{b}$ by using elementary row operations on the augmented matrix to reach row echelon (upper-triangular) form, then solving by back-substitution.

List the three elementary row operations used in Gaussian elimination.

(1) Swap two rows; (2) multiply a row by a non-zero scalar; (3) add a scalar multiple of one row to another. All preserve the solution set.

What is a pivot in Gaussian elimination, and what does the number of pivots equal?

A pivot is the first non-zero entry in a row of the echelon form, used to eliminate entries below it. The number of pivots equals $\operatorname{rank}(A)$.

How does Gauss–Jordan elimination differ from basic Gaussian elimination?

Gauss–Jordan continues reducing to reduced row echelon form (RREF): pivots are made $1$ and all entries above and below each pivot are made $0$, giving the solution directly without back-substitution.

What is partial pivoting in Gaussian elimination and why is it used?

At each step, swap rows so the pivot is the entry of largest absolute value in the current column. It is used to avoid division by zero (or tiny pivots) and to improve numerical stability against round-off error.