What is the rank of the 3x3 matrix $A =$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}$?
Rows 2 and 3 are scalar multiples of row 1, so the rank is 1.
For a linear system of equations Ax = b with A being an n x n matrix, a UNIQUE solution exists if and only if:
Ax = b has a unique solution iff A is non-singular (invertible), which is equivalent to $\det(A) != 0$.
If A is an 3 x 4 matrix with Rank(A) = 2, what is the dimension of the Null Space (Nullity) of A?
Rank-nullity gives nullity = 4 - 2 = 2 because A has 4 columns.
If A is an orthogonal matrix ($A^{T}$ * $A = I$), what are the possible values for $\det(A)$?
$\det(A^{T} \times A) = \det(A^{T}) * \det(A) = \det(A)^2 = \det(I) = 1 => \det(A) = \pm 1.$
What is the inverse of the 2x2 matrix $A =$$\begin{bmatrix} 2 & 0 \\ 0 & 5 \end{bmatrix}$?
Inverse of a diagonal matrix is obtained by taking inverses of diagonal entries: 1/2 = 0.5 and 1/5 = 0.2.