Question

Consider a transformation T: R³ → R² where R³ and R² represent three and two-dimensional real column vectors respectively. Also, T(x) = Ax for some matrix A and for each x in R³. How many rows and columns do A have and what is it's maximum possible rank?
1. Rows: 3; Columns : 2; Rank : 3
2. Rows : 3; Columns : 2; Rank : 2
3. Rows : 2; Columns : 3; Rank : 2
4. Rows : 2; Columns : 3; Rank : 3

Answer 1

Correct Answer - Option 3 : Rows : 2; Columns : 3; Rank : 2

Order of R³ = 3 × 1

Order of R² = 2 × 1

Given that:

T(x) = Ax where x ϵ R³

Let the order of A be: m × n

Since A and x matrices are multiplying, n = 3

From the above options m = 2

So order of A :  2 × 3

Rank of A = min(2, 3) = 2