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Linear Algebra – The Determinant

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    Study Guide Linear Algebra – The Determinant 1. Cramer’s Rule 1. Understanding the 2 × 2 Linear System Let’s begin with a simple system of two equations with two unknowns: Our goal is to find the values of x and y . 2. Solving by Elimination (Step - by - Step Idea) To solve for x , we eliminate y : • Multiply the first equation by (a 22 ) • Multiply the second equation by (a 12 ) • Subtract the equations This removes y and allows us to solve for x . After simplifying, we get: Similarly, to solve for y , we eliminate x : • Multiply the first equation by (a 21 ) • Multiply the second equation by (a 11 ) • Subtract This gives:

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    Study Guide Important Condition All of this works only if This expression is called the determinant of the coefficient matrix. If the determinant equals zero , the system does not have a unique solution. 3. Writing the Solution Using Determinants Now here’s where things get elegant. The denominator is actually the determinant of the coefficient matrix . So we can rewrite the solutions using determinants: This idea is called Cramer’s Rule for a 2 × 2 system. 4. Matrix Form of the System If we write the system in matrix form:

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    Study Guide Then: • The denominator is det(A) . • The numerator is the determinant of a matrix formed by replacing one column of A with B. This is the core idea of Cramer’s Rule . Extending to 3 × 3 Systems Now let’s extend this idea to three variables: If the determinant of matrix A is not zero, then the system has a unique solution . For any variable ( x j ) , the formula is: Where: • det (A) = determinant of coefficient matrix • (A j ) = matrix formed by replacing column j of A with the constants vector b 5. Two Important Theorems Cramer’s Rule leads to two important results: Theorem F has a unique solution for every b if and only if Theorem G A homogeneous system

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    Study Guide has only the trivial solution (x = 0) if and only if 6. Is Cramer’s Rule Always Practical? Not really. Although Cramer’s Rule is theoretically important , it is not efficient for large systems because computing large determinants takes time. For big systems, Gaussian elimination is usually better. However, Cramer’s Rule is very helpful when: • You only need one variable • The system is small (2×2 or 3×3) • You want a formula - based method Example 1 : Finding y Using Cramer’s Rule We are given the system: Step 1: Write in Matrix Form Now we apply Cramer’s Rule.

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    Study Guide We want y , so we: • Replace the second column of A with constants • Compute the determinant • Divide by det(A) Step 2: Compute Determinants Using row reduction and Laplace expansion: • Numerator determinant = − 123 • Denominator determinant = − 41 Step 3: Final Answer So the value of y is 3 . 2. The Classical Adjoint of a Square Matrix 1. What Is the Classical Adjoint? Let’s begin with a square matrix: The classical adjoint (also called the adjugate) of matrix (A) is defined as: The transpose of the matrix of cofactors of (A). In simple words: 1. First, find the cofactor of every entry in the matrix.

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    Study Guide 2. Arrange those cofactors into a new matrix. 3. Then take the transpose of that matrix. That final matrix is called: Example 1 : Finding the Adjoint of a Matrix We are given the matrix: Let’s find its adjoint step by step. Step 1: Find All Cofactors To find the adjoint, we must compute the cofactor 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 Matrix Now take the transpose of the cofactor matrix. This gives us:

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    Study Guide That is the classical 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 Guide 5. Finding the Inverse Using the Adjoint Now we reach a powerful formula. If: then dividing both sides by ( det A) gives: This formula works only if 6. Important Theorem (Theorem H) We can now state an important result: Theorem H A square matrix (A) is invertible if and only if: And when it is invertible, its inverse is: 7. What If the Determinant Is Zero? If: then: • The matrix is called singular

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    Study Guide • The matrix is not invertible • No inverse exists If: then: • The matrix is non - singular • The matrix is invertible An Important Result About the Adjoint We 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 something powerful: 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 ( det A) 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 Guide The exponent always depends on the size of the matrix. So the adjoint reflects whether the original matrix is singular or nonsingular. Example 2 : The Adjoint of the Adjoint What Are We Trying to Show? We want to prove something interesting: • If (A) is an invertible 2 × 2 matrix, then • But this is not always true for larger matrices. Let’s understand why. Step 1: Start with a Known Formula We already know this important identity: From this, we can rewrite: This formula connects the adjoint and the inverse. Step 2: Take the Adjoint Again Now take the adjoint of both sides:

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