NDA · Chapter

Matrices & Determinants

The concepts NDA needs, one connected idea at a time.

0 / 25

concepts mastered

Determinants0/6
Matrices0/7
Transformations0/2
Adjoint0/5
Linear equations0/4
Matrix polynomials0/1

Ready for you

Up nextValue of a determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$ ;
Elementary transformationsStartInterchange two rows, multiply a row by $k \ne 0$ , add $k$ times one row to another; the same three for columns
Minors and cofactorsStart $M_{ij}$ deletes row $i$ and column $j$ ; $A_{ij} = (-1)^{i+j}M_{ij}$ ;
Singular and invertible matricesStart $A$ has an inverse exactly when $|A| \ne 0$ ; a matrix with $|A| = 0$

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How the ideas build on each other. Each one opens once you've mastered what it needs.

Determinants0 / 6

Value of a determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$ ;

a

3 \times 3

expands along a row with signs

+ - +

Start

expanding along a row gives

|A|

is singular

a

3 \times 3

expands along a row with signs

+ - +

Start

Minors and cofactorsCofactor

A_{ij} = (-1)^{i+j}M_{ij}

: the minor with the checkerboard signLocked

Properties of determinantsSwap two rows: sign flips. Two equal rows: 0. Add a multiple of a row: no change. Factor out of one row onlyLocked

Cramer's rule

x = \frac{D_x}{D}

,

y = \frac{D_y}{D}

,

z = \frac{D_z}{D}

, with

D \ne 0

Locked

Consistency of three equationsThree lines

a_ix + b_iy = c_i

meet in a point only if the determinant of their coefficients and constants is

0

Locked

Area of a triangle and collinearityArea

= \frac{1}{2}\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|

;

0

means the points are collinearLocked

Matrices0 / 7

Types of matricesOrder is rows × columns; square, diagonal, scalar, identity, triangular, zero; singular when the determinant is 0Locked
Adding matrices and scalar multiplesSame order only; add entry by entry; a scalar multiplies every entryLocked
Multiplying matrices $(AB)_{ij}$ = row $i$ of $A$ times column $j$ of $B$ ; needs columns of $A$ = rows of $B$ Locked
Properties of matrix multiplication $AB \ne BA$ in general; $AB = O$ without $A$
The transposeRows become columns; $(A + B)^T = A^T + B^T$
Symmetric and skew-symmetric matrices $A^T = A$ or $A^T = -A$ (diagonal all 0); every square matrix is
Special square matricesOrthogonal ( $AA^T = I$ ), idempotent ( $A^2 = A$ ),

Transformations0 / 2

Elementary transformationsInterchange two rows, multiply a row by $k \ne 0$ , add $k$ times one row to another; the same three for columnsStart
Inverse by elementary transformationsWrite $A = IA$ and apply row operations until $I = BA$ (or $A = AI$ with column operations); never mix rows and columnsLocked

Adjoint0 / 5

Minors and cofactors $M_{ij}$ deletes row $i$ and column $j$ ; $A_{ij} = (-1)^{i+j}M_{ij}$ ; expanding along a row gives $|A|$ Start
Adjoint of a matrix $\operatorname{adj}A$ is the transpose of the cofactor matrix; for $2 \times 2$ , swap the diagonal and negate the other twoLocked
Singular and invertible matrices $A$ has an inverse exactly when $|A| \ne 0$ ; a matrix with $|A| = 0$
Inverse by the adjoint method $A^{-1} = \dfrac{1}{|A|}\operatorname{adj}A$ ;
Properties of the inverse and adjoint $A(\operatorname{adj}A) = |A|I$ ; $(AB)^{-1} = B^{-1}A^{-1}$ ;

Linear equations0 / 4

Matrix equations $AX = B$ gives $X = A^{-1}B$ , not $BA^{-1}$ : the inverse goes on the side where $A$ wasLocked
Solving equations by inversionWrite the system as $AX = B$ , find $A^{-1}$ , then $X = A^{-1}B$
Solving equations by reductionRow-reduce the system to a triangular form, then solve from the last equation upwardsLocked
Word problems as linear systemsName the unknowns, write one equation per condition, solve by inversion and answer in the problem's unitsLocked

Matrix polynomials0 / 1

The Cayley–Hamilton theoremA $2 \times 2$ matrix satisfies $A^2 - (\operatorname{tr}A)A + |A|I = O$ ; multiply through by $A^{-1}$ to get the inverseLocked

or

B

being

O

;

A^2

means

A \cdot A

Locked

and

(AB)^T = B^TA^T

,

order reversed

Locked

\frac{A + A^T}{2} + \frac{A - A^T}{2}

Locked

involutory

(

A^2 = I

)

and nilpotent

(

A^k = O

)

matrices; rotation matrices add their angles when multiplied

Locked

is singular

Start

check with

AA^{-1} = I

Locked

|\operatorname{adj}A| = |A|^{n-1}

;

(A^{-1})^{-1} = A

Locked

gives every unknown at once

Locked