Properties of determinants

WHAT is a determinant (so we know what we're proving)

WHY this definition? It's the only formula that is (a) linear in each row and (b) zero when two rows are equal — and those two facts force every property below.

HOW each property follows (derive, don't memorize)

Write $A$ with rows $R_1,R_2,\dots,R_n$ . Think of $\det$ as a function $D(R_1,\dots,R_n)$ .

P1. Transpose leaves it unchanged

$\det(A^\top)=\det(A)$ Why: In the permutation sum, $a_{i\sigma(i)}$ vs $a_{\sigma(i)\,i}$ — reindexing by $\sigma^{-1}$ gives the same set of terms with the same sign (since $\operatorname{sgn}(\sigma)=\operatorname{sgn}(\sigma^{-1})$ ). Consequence: every row property is also a column property.

P2. Swapping two rows flips the sign

$D(\dots,R_i,\dots,R_j,\dots)=-D(\dots,R_j,\dots,R_i,\dots)$ Why: A swap composes each permutation with a transposition, flipping $\operatorname{sgn}$ of every term.

P3. Two equal rows ⟹ determinant $=0$

Why: Swap the two equal rows: value must equal $-$ (itself). So $\det=-\det\Rightarrow\det=0$ .

P4. Multiplying one row by $k$ multiplies the determinant by $k$

$D(\dots,kR_i,\dots)=k\,D(\dots,R_i,\dots)$ Why: Every term contains exactly one factor from row $i$ , so $k$ factors out of all of them. Corollary: $\det(kA)=k^n\det A$ (each of $n$ rows contributes a $k$ ).

P5. Additivity in a single row (linearity)

$D(\dots,R_i+R_i',\dots)=D(\dots,R_i,\dots)+D(\dots,R_i',\dots)$ Why: Each term is linear in the one entry it takes from row $i$ .

P6. Adding a multiple of one row to another leaves it unchanged

$D(\dots,R_i+kR_j,\dots,R_j,\dots)=D(\dots,R_i,\dots,R_j,\dots)$ Why: By P5 it splits into the original plus $k\cdot D(\dots,R_j,\dots,R_j,\dots)$ ; the second has two equal rows, so it's $0$ (P3). This is the workhorse for hand computation.

P7. A row of zeros ⟹ determinant $=0$

Why: Take $k=0$ in P4.

P8. Triangular matrix ⟹ product of diagonal

$\det=a_{11}a_{22}\cdots a_{nn}$ Why: In a triangular matrix, the only permutation that picks a nonzero entry from every row is the identity (any off-diagonal choice forces a zero factor).

P9. Product rule

$\det(AB)=\det(A)\,\det(B)$ Why (sketch): Both sides are multilinear-alternating in the rows of $A$ and normalized on $I$ ; the axioms pin down a unique such function, forcing equality. Consequence: $\det(A^{-1})=\dfrac{1}{\det A}$ .

Worked Examples

Common mistake Steel-manned common errors

Mistake A: " $\det(A+B)=\det A+\det B$ ." Why it feels right: linearity (P5) looks like it should apply to the whole matrix. The fix: P5 is linearity in one row at a time, with all other rows fixed. Adding whole matrices changes every row simultaneously, so it doesn't factor. Test: $A=B=I_2$ gives $\det(2I)=4\neq 1+1$ .

Mistake B: " $\det(kA)=k\det A$ ." Why it feels right: P4 pulls out $k$ from a row. The fix: $kA$ scales all $n$ rows, so $\det(kA)=k^n\det A$ .

Mistake C: Row-swapping during simplification but forgetting the sign flip (P2). Fix: Every swap multiplies the running answer by $-1$ . Track it.

Recall Feynman: explain it to a 12-year-old

Imagine a stretchy sheet of graph paper. A matrix bends and stretches it. The determinant is the number that says "how many times bigger did one little square get?" If you flip the sheet over (mirror image), the number goes negative — that's the swap-rows sign flip. If two directions point the same way, the square gets squashed flat to zero area — that's "two equal rows give 0." Adding one row's worth of another row is like sliding the top of a stack of cards sideways: the shape leans but the area stays the same — so the determinant doesn't change.

Flashcards

What geometric quantity does a determinant measure?

The (signed) scaling factor of area/volume under the linear map; sign encodes orientation.

Why does swapping two rows negate the determinant?

A swap is one transposition, flipping the sign of every permutation term. (P2)

If two rows are equal, why is det

=0

Swapping the equal rows gives

\det=-\det

, so

\det=0

. (P3)

Effect of multiplying ONE row by

k

Determinant is multiplied by

k

. (P4)

Effect of multiplying the WHOLE

n\times n

matrix by

k

\det(kA)=k^n\det A

Which row operation leaves the determinant unchanged?

Adding a multiple of one row to another:

R_i\to R_i+kR_j

. (P6)

Why does

\det(A^\top)=\det A

Reindexing the permutation sum by

\sigma^{-1}

gives identical signed terms. (P1)

Determinant of a triangular matrix?

Product of the diagonal entries. (P8)

State the product rule and its corollary for inverses.

\det(AB)=\det A\det B

; hence

\det(A^{-1})=1/\det A

\det(A+B)=\det A+\det B

true?

No — linearity holds only per-row with other rows fixed.

Fastest way to compute a numeric determinant by hand?

Use P6 to make a triangular matrix, then multiply the diagonal (P8).

Connections

Determinant — Definition and Expansion by Minors
Cofactors and Adjoint of a Matrix
Inverse of a Matrix via Adjoint
Cramer's Rule and Systems of Linear Equations
Elementary Row Operations & Rank
Area of a Triangle using Determinants
Linear Transformations & Scaling of Area

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, determinant ka matlab sirf ek number nikaalna nahi hai — yeh number batata hai ki matrix apne saath jo transformation karti hai usme ek chhota sa square (area 1) kitne guna bada ya chhota ho jaata hai. Agar orientation ulti ho jaaye (mirror image), toh determinant negative ho jaata hai. Isliye do rows swap karo toh sign flip hota hai (P2), aur agar do rows bilkul same hain toh area collapse hoke line ban jaati hai, matlab determinant zero (P3). Yeh geometry samajh loge toh saari properties automatically yaad ho jaayengi.

Sabse kaam ki property exam ke liye hai P6: kisi row me doosri row ka multiple add karo, determinant bilkul nahi badalta. Iska use karke hum column me zeros bana dete hain aur matrix ko triangular bana dete hain — phir P8 se seedha diagonal multiply kar do, bas jawaab mil gaya. Bade-bade expansion se bachne ka yahi 80/20 trick hai.

Do galtiyaan bahut common hain. Pehli: log sochte hain $\det(A+B)=\det A+\det B$ — galat! Linearity sirf ek row ke liye hoti hai, poori matrix ke liye nahi. Doosri: $\det(kA)=k\det A$ likh dete hain, jabki har row scale hoti hai isliye answer $k^n\det A$ hota hai. In dono ko yaad rakho.

Mantra simple hai: "Swap Flips, Copy Kills, Scale Scales, Slide Saves." Row swap sign flip karta hai, equal (copy) row kill kar deta hai (0), row ko $k$ se scale karo toh det bhi × $k$ , aur ek row ka multiple slide karke add karo toh det save reh jaata hai. Bas itna, aur tum properties ke master ho.

WHAT is a determinant (so we know what we're proving)

HOW each property follows (derive, don't memorize)

P1. Transpose leaves it unchanged

P2. Swapping two rows flips the sign

P3. Two equal rows ⟹ determinant $=0$

P4. Multiplying one row by $k$ multiplies the determinant by $k$

P5. Additivity in a single row (linearity)

P6. Adding a multiple of one row to another leaves it unchanged

P7. A row of zeros ⟹ determinant $=0$

P8. Triangular matrix ⟹ product of diagonal

P9. Product rule

Worked Examples

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Matrices & Determinants — Introduction

Connections