JEE Main 2022 · Mathematics · Matrices and Determinants · hard

The number of matrices beginbmatrixa&b\c&dendbmatrix whose entries a,b,c,d all belong to \0,1,2\ and whose determinant is 0 is

Numerical answer

The number of matrices

\begin{bmatrix}a&b\\c&d\end{bmatrix}

whose entries

a,b,c,d

all belong to

\{0,1,2\}

and whose determinant is

0

Show answer and solution

Answer: 31

The determinant vanishes exactly when

ad=bc

. Count ordered pairs

(a,d)

by their product: product

0

occurs for

9-4=5

pairs, product

1

for

(1,1)

only, product

2

for

(1,2),(2,1)

, and product

4

for

(2,2)

. The pairs

(a,d)

and

(b,c)

must have equal products, so the number of matrices is

5^2+1^2+2^2+1^2=25+1+4+1=31

. A brute-force check over all

81

matrices confirms

31

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