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Determinants. Cramer’s Rule Part 5. Fundamentals of Engineering Analysis. Eng. Hassan S. Migdadi. Properties of Determinants. What is the determinant of a triangular matrix? How do elementary row operations effect the value of the determinant?

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## Fundamentals of Engineering Analysis

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**Determinants. Cramer’s Rule**Part 5 Fundamentals of Engineering Analysis Eng. Hassan S. Migdadi**Properties of Determinants**What is the determinant of a triangular matrix? How do elementary row operations effect the value of the determinant? What is the determinant of an elementary matrix? What is the determinant of an invertible matrix?**What is the determinant of a triangular matrix?**Hint: Expand on column 1**Row Operations**Multiply a row by a non zero constant. What happens to the determinant?**Row Operations: Add a multiple of one row to another**Hint: Expand on Row 1**Theorem 1**Multiplication of a row by a constant multiplies the determinant by that constant. Switching two rows changes the sign of the determinant. Replacing one row by that row plus a multiple of another row has no effect on the determinant.**Example – Find |A|**Strategy – Perform row operations to obtain an upper triangular matrix. Label each matrix with a new letter.**Suppose a matrix A is not invertible.**What can we say about det A? Why?**Theorem 2: A is invertible iff detA≠0.**Note – This theorem links the determinant to the invertible matrix theorem. For instance, if the columns (or rows) of A are linearly dependent, then detA=0. So if you perform row operations so that two rows or columns are the same, then detA=0.**Proof (outline)**A is invertible iff A is row equivalent to In. iff detA≠0 Note that each row operation changes the determinant by some non zero factor. Since det In=1, we couldn’t have started with a determinant of 0.**Theorem 3 – If A is an nxn matrix,**detAT=detA Proof: By induction. Theorem is obvious for n=1. Suppose it is true for n=k. Let n=k+1. The cofactor of a1j in A equals the cofactor of aj1 in AT because the cofactors involve kxk determinants and we’ve assumed the theorem is true for n=k. So the cofactor expansion along the first row of A equals the cofactor expansion along the first column of AT. By the principle of induction, the theorem is true for all n≥1.**Theorem 4 – If A and B are nxn matrices, then**detAB = (detA)(detB) Note - det(A+B)≠detA+detB

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