Matrices formulas
Master Matrices through 34 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.
Matrices, every formula
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Order of a matrix
m rows, n columns
Q1NumericalOrderThe number of elements in a matrix of order $3\times4$ is:Diagonal matrix power
Q1MCQDiagonal powerIf $A=\operatorname{diag}(2,3)$, then $A^{2}$ is:- A$\operatorname{diag}(4,9)$
- B$\operatorname{diag}(2,3)$
- C$\operatorname{diag}(4,6)$
- D$\operatorname{diag}(6,6)$
- A
Trace
sum of diagonal elements
Q1NumericalTraceThe trace of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$ is:Trace linearity
Q1NumericalTrace linearityIf $\operatorname{tr}A=5$, then $\operatorname{tr}(3A)$ is:Trace of product
Q1MCQTrace of productFor square matrices, $\operatorname{tr}(AB)$ equals:- A$\operatorname{tr}(BA)$
- B$\operatorname{tr}A\cdot\operatorname{tr}B$
- C$\operatorname{tr}A+\operatorname{tr}B$
- D$0$
- A
Matrix product entry
A is m×p, B is p×n
Q1NumericalProduct entryIf $A=[1\ 2]$ and $B=\begin{bmatrix}3\\4\end{bmatrix}$, the single entry of $AB$ is:Conformability
columns of A = rows of B
Q1MCQConformabilityThe product $A_{2\times3}B_{3\times4}$ has order:- A$2\times4$
- B$3\times3$
- C$4\times2$
- Dundefined
- A
Non-commutativity
Q1MCQNon-commutativityFor general square matrices $A,B$:- A$AB\neq BA$ in general
- B$AB=BA$ always
- C$AB=O$ always
- D$AB=A+B$
- A
(A+B)^2
Q1MCQ(A+B)^2$(A+B)^{2}$ equals:- A$A^{2}+B^{2}+AB+BA$
- B$A^{2}+2AB+B^{2}$
- C$A^{2}+B^{2}$
- D$A^{2}+2BA+B^{2}$
- A
Transpose of product
Q1MCQTranspose of product$(AB)^{T}$ equals:- A$B^{T}A^{T}$
- B$A^{T}B^{T}$
- C$(BA)^{T}$
- D$AB$
- A
Transpose of sum
Q1MCQTranspose of sum$(A+B)^{T}$ equals:- A$A^{T}+B^{T}$
- B$B^{T}A^{T}$
- C$A^{T}-B^{T}$
- D$(AB)^{T}$
- A
Symmetric matrix
Q1MCQSymmetricA square matrix $A$ is symmetric iff:- A$A^{T}=A$
- B$A^{T}=-A$
- C$A^{2}=A$
- D$A^{2}=I$
- A
Skew-symmetric matrix
Q1NumericalSkew-symmetric traceThe trace of any skew-symmetric matrix is:Symmetric + skew decomposition
unique sym + skew split
Q1MCQSym+skew splitEvery square matrix $A$ can be written as:- A$\tfrac12(A+A^{T})+\tfrac12(A-A^{T})$
- B$A+A^{T}$
- C$A\cdot A^{T}$
- D$\tfrac12(A+A^{T})$
- A
Orthogonal matrix
Q1MCQOrthogonalIf $A$ is orthogonal, then $A^{-1}$ equals:- A$A^{T}$
- B$A$
- C$-A$
- D$\operatorname{adj}A$
- A
Idempotent matrix
Q1MCQIdempotentA matrix satisfying $A^{2}=A$ is called:- Aidempotent
- Binvolutory
- Cnilpotent
- Dorthogonal
- A
Involutory matrix
Q1MCQInvolutoryA matrix satisfying $A^{2}=I$ is called:- Ainvolutory
- Bidempotent
- Cnilpotent
- Dsingular
- A
Nilpotent matrix
Q1MCQNilpotentA matrix $A$ with $A^{p}=O$ for some $p$ is called:- Anilpotent
- Bidempotent
- Corthogonal
- Dinvolutory
- A
Determinant 2×2
Q1NumericalDeterminant 2×2The value of $\begin{vmatrix}2&3\\1&4\end{vmatrix}$ is:Cofactor
M_{ij} = minor
Q1NumericalCofactorFor a $3\times3$ matrix, the sign factor $(-1)^{i+j}$ for the $(2,3)$ cofactor is (enter $+1$ or $-1$):Expansion along a row
Q1MCQRow expansionThe determinant of a $3\times3$ matrix along row $i$ is:- A$a_{i1}A_{i1}+a_{i2}A_{i2}+a_{i3}A_{i3}$
- B$a_{i1}M_{i1}+a_{i2}M_{i2}+a_{i3}M_{i3}$
- C$a_{i1}+a_{i2}+a_{i3}$
- D$A_{i1}A_{i2}A_{i3}$
- A
Determinant of scalar multiple
n = order of A
Q1Numerical|kA|If $A$ is $3\times3$ with $|A|=2$, then $|3A|$ is:Determinant of product
Q1Numerical|AB|If $|A|=3$ and $|B|=4$, then $|AB|$ is:Determinant of transpose
Q1MCQ|A^T|$|A^{T}|$ equals:- A$|A|$
- B$-|A|$
- C$\dfrac{1}{|A|}$
- D$|A|^{2}$
- A
A·adj A
Q1MCQA·adj A$A(\operatorname{adj}A)$ equals:- A$|A|\,I$
- B$I$
- C$A^{2}$
- D$O$
- A
Determinant of adjoint
Q1Numerical|adj A|If $A$ is $3\times3$ with $|A|=2$, then $|\operatorname{adj}A|$ is:adj of adj
|A|≠0
Q1MCQadj(adj A)For a $3\times3$ matrix, $\operatorname{adj}(\operatorname{adj}A)$ equals:- A$|A|\,A$
- B$A$
- C$A^{2}$
- D$|A|^{2}A$
- A
Inverse via adjoint
|A|≠0
Q1MCQInverse via adj$A^{-1}$ equals:- A$\dfrac{\operatorname{adj}A}{|A|}$
- B$\dfrac{|A|}{\operatorname{adj}A}$
- C$|A|\operatorname{adj}A$
- D$A^{T}$
- A
Inverse of product
Q1MCQInverse of product$(AB)^{-1}$ equals:- A$B^{-1}A^{-1}$
- B$A^{-1}B^{-1}$
- C$(BA)^{-1}$
- D$AB$
- A
Determinant of inverse
Q1Numerical|A^{-1}|If $|A|=5$, then $|A^{-1}|$ is (enter as a decimal):Characteristic equation (3×3)
S1=tr, S3=|A|
Q1MCQCharacteristic equationThe characteristic equation of a $3\times3$ matrix is:- A$\lambda^{3}-S_1\lambda^{2}+S_2\lambda-S_3=0$
- B$\lambda^{3}+S_1\lambda^{2}+S_2\lambda+S_3=0$
- C$\lambda^{2}-S_1\lambda+S_2=0$
- D$\lambda^{3}-S_3=0$
- A
Cramer's rule
Δ≠0
Q1MCQCramer's ruleBy Cramer's rule, $x$ equals:- A$\dfrac{\Delta_1}{\Delta}$
- B$\dfrac{\Delta}{\Delta_1}$
- C$\Delta_1\Delta$
- D$\Delta-\Delta_1$
- A
Homogeneous non-trivial solution
Q1MCQHomogeneous system$AX=O$ has a non-trivial solution iff:- A$|A|=0$
- B$|A|\neq0$
- C$A=I$
- D$A$ is orthogonal
- A
Vandermonde determinant
Q1MCQVandermonde$\begin{vmatrix}1&a&a^{2}\\1&b&b^{2}\\1&c&c^{2}\end{vmatrix}$ equals:- A$(a-b)(b-c)(c-a)$
- B$(a+b)(b+c)(c+a)$
- C$abc$
- D$a^{2}+b^{2}+c^{2}$
- A
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