Matrix Multiplication Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Compute [20−1132][14−1]\begin{bmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 4 \\ -1 \end{bmatrix}.

Solution

  1. 1
    Step 1: Dimensions: (2×3)(3×1)→2×1(2 \times 3)(3 \times 1) \to 2 \times 1.
  2. 2
    Step 2: Row 1: 2(1)+0(4)+(−1)(−1)=2+0+1=32(1) + 0(4) + (-1)(-1) = 2 + 0 + 1 = 3.
  3. 3
    Step 3: Row 2: 1(1)+3(4)+2(−1)=1+12−2=111(1) + 3(4) + 2(-1) = 1 + 12 - 2 = 11.
  4. 4
    Result: [311]\begin{bmatrix} 3 \\ 11 \end{bmatrix}

Answer

[311]\begin{bmatrix} 3 \\ 11 \end{bmatrix}
Multiplying a matrix by a column vector produces another column vector. Each entry is the dot product of a row of the matrix with the column vector. This is the core operation in systems of linear equations Ax=bAx = b.

About Matrix Multiplication

Multiplying matrices AA (m×nm \times n) and BB (n×pn \times p) by taking dot products of rows of AA with columns of BB to produce an m×pm \times p result.

Learn more about Matrix Multiplication →

More Matrix Multiplication Examples

Example 1 medium

Compute [formula].

Example 3 easy

Can you multiply a [formula] matrix by a [formula] matrix? Why or why not?

Example 4 medium

Compute [formula].

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