Linear Algebra – The Determinant - Quick Guides

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Study GuideLinear Algebra–The Determinant1.Cramer’s Rule1. Understanding the 2×2 Linear SystemLet’s begin with a simple system of two equations with two unknowns:Our goal is to find the values ofxandy.2. Solving byElimination (Step-by-Step Idea)To solve forx, we eliminatey:•Multiply the first equation by (a22)•Multiply the second equation by (a12)•Subtract the equationsThis removesyand allows us to solve forx.After simplifying, we get:Similarly, to solve fory, we eliminatex:•Multiply the first equation by (a21)•Multiply the second equation by (a11)•SubtractThis gives:

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Study GuideImportant ConditionAll of this worksonly ifThis expression is called thedeterminantof the coefficient matrix.If the determinant equalszero, the system does not have a unique solution.3. Writing the Solution Using DeterminantsNow here’s where things get elegant.The denominatoris actually thedeterminant of thecoefficient matrix.So we can rewrite the solutions using determinants:This idea is calledCramer’s Rulefor a 2×2 system.4. Matrix Form of the SystemIf we write the system in matrix form:

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Study GuideThen:•The denominator isdet(A).•The numerator isthe determinant of a matrix formed by replacing one column of A with B.This is the core idea ofCramer’s Rule.Extending to 3×3 SystemsNow let’s extend this idea to three variables:If the determinant of matrixAis not zero, then the system has aunique solution.For any variable(xj), the formula is:Where:•det(A) = determinant of coefficient matrix•(Aj) = matrix formed by replacing columnjof A with the constants vectorb5. Two ImportantTheoremsCramer’s Rule leads to two important results:Theorem Fhas aunique solution for every bif and only ifTheorem GA homogeneous system

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Study Guidehas only thetrivial solution (x = 0)if and only if6. Is Cramer’s Rule Always Practical?Notreally.Although Cramer’s Rule istheoretically important, it is not efficient for large systems becausecomputing large determinants takes time.For big systems,Gaussian eliminationis usually better.However, Cramer’s Rule is very helpful when:•Youonly need one variable•The system is small (2×2 or 3×3)•You want a formula-based methodExample1:Finding y Using Cramer’s RuleWe are given the system:Step 1:Write in Matrix FormNow we apply Cramer’s Rule.

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Study GuideWe wanty, so we:•Replace thesecond column of A with constants•Compute the determinant•Divide by det(A)Step 2:Compute DeterminantsUsing row reduction and Laplace expansion:•Numerator determinant =−123•Denominator determinant =−41Step 3:Final AnswerSo the value ofy is 3.2.The Classical Adjoint of a Square Matrix1. What Is the Classical Adjoint?Let’s begin with a square matrix:Theclassical adjoint(also called the adjugate) of matrix (A) is defined as:The transpose of the matrix ofcofactors of (A).In simple words:1.First, find thecofactorof every entry in the matrix.

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Study Guide2.Arrange those cofactors into a new matrix.3.Then take thetransposeof that matrix.That final matrix is called:Example1:Finding the Adjoint of a MatrixWe are given the matrix:Let’s find its adjoint step by step.Step 1:Find All CofactorsTo find the adjoint, we must compute thecofactor of every entry.Remember:So for each entry:•Remove its row and column.•Compute the determinant of what remains.•Apply the sign pattern (+ − + / − + − / + − +).After calculating all cofactors, we form the cofactor matrix.Step 2:Transpose the Cofactor MatrixNow take the transpose of the cofactor matrix.This gives us:

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Study GuideThat is theclassical adjoint of A.3. Why Do We Form the Adjoint?Now comes the important question:Why is the adjoint useful?Let’s multiply the matrix by its adjoint:After performing the multiplication, we obtain:where (I) is the identity matrix.This result is extremely important.4. What Does This Tell Us?Notice something special:So when we calculate the determinant of (A), we get:Therefore,This relationship always holds for square matrices.

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Study Guide5. Finding the Inverse Using theAdjointNow we reach a powerful formula.If:then dividing both sides by (detA) gives:This formula worksonly if6. Important Theorem (Theorem H)We can now state an important result:Theorem HA square matrix (A) isinvertibleif and only if:And when it is invertible, its inverse is:7. What If the Determinant Is Zero?If:then:•The matrix is calledsingular

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Study Guide•The matrix isnot invertible•No inverse existsIf:then:•The matrix isnon-singular•The matrixis invertibleAnImportant Result About the AdjointWe now arrive at a very useful and elegant result:where:•(A) is an (n n) matrix•(A) is the classical adjoint of (A)•(det A) is the determinant of (A)What Does This Mean?This formula tells us somethingpowerful:If you know the determinant of a matrix (A), then you can easily find the determinant of its adjoint.Instead of recomputing everything from scratch, you simply:1.Take (detA)2.Raise it to the power (n-1)That’s it!Why Is the Power(n-1)?If the matrix is:

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Study GuideThe exponent always depends on the size of the matrix.So the adjoint reflects whether the original matrix is singular or nonsingular.Example2:The Adjoint of the AdjointWhat Are We Trying to Show?We want to provesomething interesting:•If (A) is an invertible2×2matrix, then•But this isnot always truefor larger matrices.Let’s understand why.Step 1:Start with a Known FormulaWe already know this important identity:From this, we can rewrite:Thisformula connects the adjoint and the inverse.Step 2:Take the Adjoint AgainNow take the adjoint of both sides:

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