Week 4 Checkpoint

A short checkpoint on Week 4: matrix shapes, the inner-dimension rule for matrix multiplication, the transpose, matrix–vector products, writing systems as $A\mathbf{x} = \mathbf{b}$, and classifying solutions as unique, infinite, or none. Aim for 80% or better before moving to Week 5. Each explanation names the chapter to revisit if you miss it.

  1. 1.

    AA has shape (2,3)(2, 3) and BB has shape (3,5)(3, 5). What is the shape of the product ABAB?

  2. 2.

    AA has shape (4,2)(4, 2) and BB has shape (4,2)(4, 2). Which product is defined?

  3. 3.

    Compute the matrix–vector product AxA\mathbf{x} for A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and x=[56]\mathbf{x} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}.

  4. 4.

    For A=[102314]A = \begin{bmatrix} 1 & 0 & 2 \\ 3 & 1 & 4 \end{bmatrix}, what is the entry in row 1, column 2 of A⊤A^\top (using 1-based indexing)?

  5. 5.

    The system   x+y=2,  x+y=5  \;x + y = 2,\; x + y = 5\; written as Ax=bA\mathbf{x} = \mathbf{b} has how many solutions?

  6. 6.

    A system Ax=bA\mathbf{x} = \mathbf{b} has two equations in three unknowns and is consistent. What is the most it can have?