Solving Systems of Linear Equations using Matrices

Study Guide

Key Definition

In a linear system $Ax = b$, the coefficient matrix $A$ contains the system coefficients, the augmented matrix $[A\,|\,b]$ combines coefficients with constants, and if $det(A) \neq 0$ then the solution is $x = A^{-1} b$.

Important Notes

A matrix is a rectangular array of numbers.
Matrices focus only on the coefficients of variables.
Matrix operations can replace traditional algebraic manipulation.
The determinant of a matrix helps in solving equations.
Matrix inversion is crucial for finding solutions.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$[A]_{ij}$ entry in row $i$, column $j$

$A^{T}$ transpose of matrix $A$

$A^{-1}$ inverse of matrix $A$ (if $det(A)
eq 0$)

$|A|$ determinant of A

Remember to use proper notation when solving problems

Why It Works

By organizing all coefficients into matrix $A$ and constants into vector $b$, we leverage matrix operations to solve systems systematically. Inverse matrices or row reduction convert solving many equations into arithmetic steps, reducing error and computational work.

Remember

Always verify that $det(A)
eq 0$ before inverting. Align rows and columns properly when forming matrices. For singular matrices ($det(A)=0$), use row reduction to determine if there are no or infinitely many solutions.

Quick Reference

Matrix Multiplication:$(AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj}, \text{ where A is m\times n and B is n\times p}$

Matrix Inversion:$A A^{-1} = I \quad\text{and if }A x = b\text{ then }x = A^{-1} b$

Determinant:$det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} \;det(M_{ij});\quad A ext{ is invertible if }det(A)
eq 0$

Understanding Solving Systems of Linear Equations using Matrices

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

For a simple 2×2 system, write $A = [[a, b], [c, d]]$ and $b = [e, f]$, compute $det(A)=ad-bc$, then $A^{-1}=1/det(A) * [[d, -b], [-c, a]]$ and $x=A^{-1}b$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the resulting matrix from $A imes B$ if $A = [[1,2],[3,4]]$ and $B = [[2,0],[1,2]]$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You want to buy shirts and pants. A shirt costs $10 and a pair of pants costs $20. You have $100 and want to buy exactly 7 items. How many shirts (x) and pants (y) can you buy?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the inverse of the given matrix: $[[2,1],[5,3]]$

Challenge Quiz

Single Choice Quiz

Advanced

Solve the system using matrices: {3x + 4y = 5; 2x - y = 7}

Recap

Watch & Learn

Review key concepts and takeaways