Vectors and Matrices

Vectors and Matrices

Vectors and matrices are short, procedural topics on the ACT. You are rarely asked for more than one or two steps, and almost every question is solved by following a formula carefully. Most lost points come from sign errors and from mixing up which operation applies.

Vectors

A vector has both magnitude and direction, and is written as an ordered pair $(x, y)$. Vectors add and subtract by combining their corresponding components separately: $(2, -3) + (-5, 1) = (-3, -2)$. Be careful with subtraction when a component is negative: $(8, 3) - (-2, 6) = (8 + 2, 3 - 6) = (10, -3)$. A scalar multiple scales both components: $2(3, -2) = (6, -4)$.

The magnitude of a vector is found with the same formula as the distance formula, $\sqrt{x^2 + y^2}$, since a vector's magnitude is its length from the origin. The magnitude of $(5, 12)$ is $\sqrt{25 + 144} = 13$. Square the components before adding; the magnitude is not $5 + 12$.

When a question combines operations, do the vector arithmetic first and apply the magnitude formula last. To find the magnitude of $u - v$ for $u = (7, 1)$ and $v = (2, 5)$, compute $u - v = (5, -4)$ and then $\sqrt{25 + 16} = \sqrt{41} \approx 6.40$.

Where this appears as a model. Two velocities combined. A boat moving at $(3, 4)$ miles per hour relative to the water, on a current moving at $(1, 0)$ miles per hour, has a resultant ground velocity of $(4, 4)$, a speed of $\sqrt{32} \approx 5.66$ miles per hour.

Matrix addition, scalar multiples, and determinants

A matrix is added, subtracted, or scaled entry by entry. Adding two matrices combines corresponding entries; multiplying a matrix by a scalar multiplies every entry by that scalar, including the negative sign if there is one. For example, $-2\begin{bmatrix} 3 & -2 \\ 5 & 4 \end{bmatrix} = \begin{bmatrix} -6 & 4 \\ -10 & -8 \end{bmatrix}$.

The determinant of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $ad - bc$, a single number used in later work such as finding an inverse or classifying a system of equations. Multiply down the main diagonal, multiply across the other diagonal, and subtract. The determinant of $\begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}$ is $(5)(4) - (2)(3) = 14$.

Matrix multiplication

Matrix multiplication is not entry by entry. Each entry of the product comes from a row of the first matrix dotted with a column of the second: multiply corresponding entries and add the results. Multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second, and the result takes the number of rows of the first matrix and the number of columns of the second. A $2 \times 3$ matrix times a $3 \times 2$ matrix is defined and gives a $2 \times 2$ result.

Worked Example. Let $A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -1 \\ 2 & 5 \end{bmatrix}$. Find $AB$.

Row 1, Column 1: $(2)(4) + (1)(2) = 10$.
Row 1, Column 2: $(2)(-1) + (1)(5) = 3$.
Row 2, Column 1: $(0)(4) + (3)(2) = 6$.
Row 2, Column 2: $(0)(-1) + (3)(5) = 15$.

So $AB = \begin{bmatrix} 10 & 3 \\ 6 & 15 \end{bmatrix}$.

Order matters. In general $AB \ne BA$, and one of the two products may not even be defined. Always check the dimensions before you start multiplying.

Systems as matrix equations

A system of two linear equations can be written as a single matrix equation. Each row of the coefficient matrix holds one equation's coefficients, in the same order as the variable column, and multiplies out to equal the constant on that equation's right side. The system $3x + 2y = 8$ and $x - y = 1$ becomes $\begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 1 \end{bmatrix}$. Check that the columns of the coefficient matrix line up with $x$ and $y$, and that the constants stay in the same order as the equations.

Matrix algebra

Worked Example. Solve $2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix}$ for the matrix $X$.

Isolate $X$ the same way you would isolate a variable, using matrix subtraction and then scalar division, entry by entry. $2X = \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 6 & 6 \\ 6 & 6 \end{bmatrix}$, so $X = \begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix}$. Check: $2\begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix} + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix}$, which matches.

The best route is hand arithmetic on each entry. A familiar approved handheld calculator with matrix functions can also do this directly; the ACT's embedded Desmos tool has no dedicated matrix mode to rely on here. The common error is subtracting the matrices in the wrong order, or forgetting to divide every entry by the scalar.

Strategy summary

• Vectors: add or subtract components; magnitude is $\sqrt{x^2 + y^2}$, applied last.
• Determinant: $ad - bc$, with a subtraction, not an addition.
• Product of matrices: check dimensions first, then row-times-column; never entry by entry.
• Scalar times a matrix: every entry, with the sign.
• Matrix equation for a system: rows are equations, columns are variables.