Matrix Basics and Operations

Matrices organize numbers in a rectangular array of rows and columns. A matrix's dimensions are written as rows × columns, such as 2×3 (2 rows, 3 columns). Each element is labeled with subscripts that show its position (row, column).

The identity matrix is a special square matrix with 1s down the diagonal and 0s everywhere else. It functions like the number 1 in multiplication. For example, a 2×2 identity matrix looks like:

[1 0]
[0 1]

Matrix addition and subtraction work by combining corresponding elements, but only matrices with identical dimensions can be added or subtracted. For example:

[3 -8]   [6  2]   [9  -6]
[0 12] + [3 -9] = [3   3]

💡 Unlike multiplication, matrix addition is commutative, meaning A + B = B + A.

Scalar multiplication happens when you multiply every element in a matrix by a number. For "division," you multiply by a fraction $like A/3 = (1/3)A$. Matrix multiplication is trickier—you must match the inner dimensions and use dot products of rows and columns. Remember that matrix multiplication is not commutative—the order matters!

Linear Systems Using Matrices

A linear system contains equations where variables appear only to the first power, without multiplying or dividing other variables, and without radicals. These systems can have one solution, no solutions, or infinitely many solutions.

To solve linear systems, we can convert them into augmented matrices that contain just the coefficients and constants. For example, the system

ax + by = c
dx + ey = f

becomes the matrix

[a b | c]
[d e | f]

We use row operations to transform these matrices:

1. Interchange rows
2. Multiply a row by a non-zero number
3. Replace a row by adding/subtracting another row
4. Combine operations to replace a row with itself plus a multiple of another row

💡 The goal is to reach reduced row echelon form, where the left side is an identity matrix, giving you the solution directly.

When solving, you might encounter a row of all zeros. If this row has a non-zero value on the right side, the system has no solutions. If the row is completely zeros, the system has infinitely many solutions, which you'll express in terms of a parameter.

For example, if your final matrix shows x = 4, y = -3, z = 1, that's your unique solution. If you end up with a relationship between variables, you'll have infinitely many solutions.

Inverse Matrices and Determinants

Only square matrices can have inverses. A matrix A and its inverse A⁻¹ multiply to give the identity matrix: A·A⁻¹ = I. To find an inverse:

1. Create an augmented matrix with your original matrix on the left and the identity matrix on the right
2. Use row operations to transform the left side into the identity matrix
3. The right side becomes your inverse matrix

Not all matrices have inverses. If you get a row of zeros during this process, the matrix is singular and has no inverse.

💡 For 2×2 matrices, there's a shortcut: if A = [a b; c d], then A⁻¹ = 1/det(A) · $d -b; -c a$ (swap the diagonal elements, negate the others, and divide by the determinant).

The determinant of a matrix (written as det A or |A|) is a special number that tells us important properties about the matrix:

• For 2×2 matrices: det [a b; c d] = ad - bc
• For 3×3 matrices: you can use diagonal products or expansion by minors

If a matrix's determinant equals zero, the matrix is singular and has no inverse. This is incredibly useful for quickly determining whether a linear system has a unique solution.