AP Precalculus: Matrices

Matrices can model a system that changes between a finite number of states at regular times, such as a customer who is subscribed or not subscribed each month. If the next state depends only on the current state, the process is a Markov chain. Its transition matrix P has entries p_ij = the probability of moving from state j to state i. Column j lists where you go from state j, so each column adds to 1; a matrix with that property is a Markov matrix. A state vector x⁽ⁿ⁾ is a column whose i-th entry is the probability of being in state i at observation n; its entries are nonnegative and sum to 1. The key rule is x⁽ⁿ⁺¹⁾ = P·x⁽ⁿ⁾, so x⁽ⁿ⁾ = Pⁿx⁽⁰⁾. Repeating the multiplication (a calculator makes this fast) reveals the steady state, a distribution that no longer changes from one step to the next. You can also look backward: since x⁽ⁿ⁾ = Px⁽ⁿ⁻¹⁾, multiplying by P⁻¹ gives x⁽ⁿ⁻¹⁾ = P⁻¹x⁽ⁿ⁾, which predicts an earlier state. Always label the rows and columns so you can tell from-state and to-state apart.

1. Matrices: Order, Elements, and Equality

A matrix is a rectangular array of numbers arranged in rows (horizontal) and columns (vertical). The numbers are its elements or entries. Matrices are everywhere: a calendar is a matrix with weeks as rows and days as columns, and a recipe card is a matrix with ingredients as rows and amounts as a column. The order (or size) of a matrix is rows × columns, read “m by n.” A matrix with one column is a column matrix (column vector), one with one row is a row matrix (row vector), and one with the same number of rows and columns is a square matrix. To name an element, always give the row first, then the column: the entry in row 2, column 3 of A is written a₂₃. Two matrices are equal only if they have the same order and every pair of corresponding entries is equal, which turns one matrix equation into one ordinary equation for each position.

2. Adding and Subtracting Matrices, and Scalar Multiples

Matrices of the same order are added or subtracted by combining corresponding entries: (A + B)_ij = a_ij + b_ij, and likewise for subtraction. If the orders differ, the sum or difference is not defined. Addition is commutative (A + B = B + A) and associative, and the zero matrix, with every entry 0, is the additive identity. A scalar multiple kA multiplies every entry of A by the number k, so 3A triples each entry and −A changes every sign. These rules let you solve simple matrix equations just as you would solve x + 5 = 9: to solve X + A = B, subtract A from both sides to get X = B − A. Check the orders before you start, and watch the signs when you subtract a negative entry.

3. Matrix Multiplication

The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p: the inner numbers must match and the outer numbers give the order of the answer. To find the entry in row i, column j of AB, take row i of A and column j of B, multiply the pairs in order, and add: (AB)_ij = a_i1b_1j + a_i2b_2j + …. Think “across, then down.” Matrix multiplication is not commutative: AB and BA can have different orders, and even when both are defined they usually have different entries. It is associative, (AB)C = A(BC), and distributes over addition, A(B + C) = AB + AC. A matrix times a column vector gives a column vector, which is the main use in the later sections. Always write down the orders first: if the inner numbers do not match, stop, because the product does not exist.

4. The Identity Matrix and Inverse Matrices

The identity matrix I is the square matrix with 1s along the main diagonal (top left to bottom right) and 0s everywhere else, for example 1001 and 100010001. It plays the role of the number 1: for any square matrix A of the same order, AI = IA = A. An n × n matrix A is invertible if there is an n × n matrix B with AB = BA = I. That matrix B is the inverse of A, written A⁻¹. Not every matrix has an inverse, and only square matrices can. To verify that two matrices are inverses, multiply them and check that the product is I; because multiplication is not commutative in general, it is good practice to check both orders (for square matrices one order implies the other). Be careful with notation: A⁻¹ does not mean 1/A or the matrix of reciprocals of the entries.

5. Finding an Inverse: Systems, the Determinant Formula, and Technology

There are three ways to find A⁻¹ for a 2 × 2 matrix. Method 1: system of equations. Let A⁻¹ = wxyz and use AA⁻¹ = I. Multiplying gives four equations that split into two small systems (one for the first column, w and y, and one for the second column, x and z), which you solve by elimination or substitution. Method 2: the determinant formula. For A = abcd, the determinant is det(A) = ad − bc. If det(A) ≠ 0, then A⁻¹ = 1det(A) d−b−ca. In words: swap the two diagonal entries, change the sign of the other two, and divide everything by the determinant. Method 3: technology. On a graphing calculator, enter the matrix in the matrix editor (2nd, x⁻¹, EDIT), set its dimensions, then return to the home screen, select [A], press x⁻¹ and ENTER; use the MATH menu to convert decimals to fractions. The key fact is that a square matrix is invertible if and only if det(A) ≠ 0. If det(A) = 0, the matrix is called singular and has no inverse.

6. The Determinant and the Area of a Parallelogram

Matrices are a handy tool for vectors. Take two vectors v₁ = ⟨a, c⟩ and v₂ = ⟨b, d⟩ drawn from the origin. Together they form two sides of a parallelogram, and the fourth vertex is the head of v₁ + v₂. Put the vectors as the two columns of A = abcd. Then the area of the parallelogram equals |det(A)|, the absolute value of ad − bc. The absolute value is needed because the determinant can be negative (it depends on the order of the vectors) but an area cannot. It works equally well if the vectors are written as rows. If det(A) = 0 the area is zero, which means the two vectors lie on the same line, so they are parallel and the “parallelogram” is flat. When a parallelogram is given by its vertices rather than vectors, pick one vertex, form the two vectors that go from it to the two adjacent vertices (head minus tail), and take the determinant of those. A graph of the points first helps you pick the adjacent vertices.

7. Functions from Rⁿ to R^m and Linear Transformations

A function f from the real numbers to the real numbers sends a number to a number. Now we allow the inputs and outputs to be ordered pairs, triples, or n-tuples. The set of all ordered n-tuples (x₁, x₂, …, x_n) of real numbers is n-space, written Rⁿ. A point (a₁, a₂) in R² can be read two ways: as a point with those coordinates, or as a vector with those components. A rule that takes each element of Rⁿ to an element of R^m is a transformation (or map) T: Rⁿ → R^m. It can be defined by m formulas, w₁ = f₁(x₁, …, x_n), …, w_m = f_m(x₁, …, x_n), and the output is the image of the input. If m = n it is called an operator. A transformation is linear when every output formula is a sum of constant multiples of the inputs, such as w = 3x − 2y + z. Squares, products of variables (xy), and added constants (x + 1) make it nonlinear. A linear transformation always sends the zero vector to the zero vector.

8. The Standard Matrix and Matrix-Vector Multiplication

Every matrix defines a linear transformation, and every linear transformation comes from a matrix. If A is m × n, then T(x) = Ax is a linear transformation from Rⁿ to R^m, where x is written as a column. For example, multiplying 10−1312 by the column (x, y, z) gives (x − z, 3x + y + 2z), a map from R³ to R². To go the other way, from formulas to the standard matrix, you can read the coefficients off row by row, but there is a faster trick: the columns of the standard matrix are the images of the unit vectors. In R², the first column is T(1, 0) and the second is T(0, 1). In general, column j is T(e_j), where e_j has a 1 in position j and 0 elsewhere. That is why the unit vectors i and j are so useful: if you know where they land, you know where everything lands, since (x, y) = x·i + y·j and T(x, y) = x·T(i) + y·T(j). The matrix of an m-output, n-input transformation is m × n, so check that the number of rows equals the number of outputs.

9. Rotation Operators

A rotation operator on R² turns every vector counterclockwise about the origin through a fixed angle θ and does not change its length. Its standard matrix comes from the unit-vector trick: i = (1, 0) rotates to (cos θ, sin θ) and j = (0, 1) rotates to (−sin θ, cos θ), so R(θ) = cos θ−sin θsin θcos θ. The image of (x, y) is therefore (x cos θ − y sin θ, x sin θ + y cos θ). A positive angle rotates counterclockwise and a negative angle rotates clockwise. Common matrices: for θ = 90°, R = 0−110; for θ = 180°, R = −100−1, which just negates both coordinates. Because rotation is a rigid motion, its determinant is cos²θ + sin²θ = 1, so areas are preserved. Use exact values for special angles (30°, 45°, 60°, 90°) and keep your calculator in the right angle mode.

10. Dilations and the Determinant

A transformation T(x) = Ax from R² to R² maps the plane to itself, and the determinant of A describes how it changes area. The unit square, with corners (0, 0), (1, 0), (1, 1), (0, 1), is the parallelogram spanned by the unit vectors i and j, and its area is 1. After the transformation it becomes the parallelogram spanned by T(i) and T(j), which are the columns of A. By the previous section, the area of that parallelogram is |det(A)|. This holds for every region, not just squares: |det(A)| is the factor by which T multiplies the area of any region. If |det(A)| > 1 the map stretches areas (an expansion or dilation); if 0 < |det(A)| < 1 it shrinks them; if det(A) < 0 the orientation is flipped as well; and if det(A) = 0 the whole plane is collapsed onto a line or a point, so the map has no inverse. A pure scaling by k in both directions has matrix k00k and det = k², so it multiplies area by k² and lengths by |k|.

11. Compositions and Inverses of Linear Transformations

If T_A: Rⁿ → R^k and T_B: R^k → R^m are linear transformations, you can apply T_A first and then T_B. This composition is written (T_B ∘ T_A)(x) = T_B(T_A(x)) = B(Ax) = (BA)x. Two big facts follow: (1) the composition of two linear transformations is a linear transformation, and (2) its matrix is the product BA, with the matrix of the second (outer) transformation on the left. Because matrix multiplication is not commutative, the order of the transformations matters. The inverse of a linear transformation undoes it. For a square matrix A, the following are equivalent: A is invertible; det(A) ≠ 0; the range of T_A is all of Rⁿ; and T_A is one-to-one (different inputs give different outputs). In that case T_A⁻¹ is multiplication by A⁻¹, and T_A⁻¹(T_A(x)) = A⁻¹Ax = x: applying the transformation and then its inverse brings you back to the start. To find a formula for the inverse, write the standard matrix, invert it, and multiply by (w₁, w₂).

12. Matrices Modeling Contexts: Markov Chains

Matrices can model a system that changes between a finite number of states at regular times, such as a customer who is subscribed or not subscribed each month. If the next state depends only on the current state, the process is a Markov chain. Its transition matrix P has entries p_ij = the probability of moving from state j to state i. Column j lists where you go from state j, so each column adds to 1; a matrix with that property is a Markov matrix. A state vector x⁽ⁿ⁾ is a column whose i-th entry is the probability of being in state i at observation n; its entries are nonnegative and sum to 1. The key rule is x⁽ⁿ⁺¹⁾ = P·x⁽ⁿ⁾, so x⁽ⁿ⁾ = Pⁿx⁽⁰⁾. Repeating the multiplication (a calculator makes this fast) reveals the steady state, a distribution that no longer changes from one step to the next. You can also look backward: since x⁽ⁿ⁾ = Px⁽ⁿ⁻¹⁾, multiplying by P⁻¹ gives x⁽ⁿ⁻¹⁾ = P⁻¹x⁽ⁿ⁾, which predicts an earlier state. Always label the rows and columns so you can tell from-state and to-state apart.

Common Mistakes to Avoid

• Reversing the order when naming matrix elements. The entry a₂₃ is in row 2, column 3, never column 2, row 3.
• Trying to add or subtract matrices of different orders. The orders must be exactly the same; multiplication has a different rule.
• Multiplying matrices entry by entry. Matrix multiplication takes row times column sums, and the inner dimensions must match.
• Assuming AB = BA. Matrix multiplication is not commutative, and in a composition the matrix of the second transformation goes on the left.
• Writing the inverse of abcd with the wrong signs or forgetting to divide by the determinant. Swap a and d, negate b and c, then divide by ad − bc.
• Forgetting that A⁻¹ exists only when det(A) ≠ 0, and using the determinant (not its absolute value) as an area.
• Building a standard matrix with the images of i and j as rows. They are the columns.
• In a Markov chain, filling the transition matrix by rows instead of columns. Each column is a from-state and must add to 1.

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