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Properties of Determinants

Properties of Determinants:
·  Let A be an n × n  matrix and c be a scalar then:


·  Suppose that A, B, and C are all n × n  matrices and that they differ by only a row, say the kth row.  Let’s further suppose that the kth row of C can be found by adding the corresponding entries from the kth rows of A and B.  Then in this case we will have:

The same result will hold if we replace the word row with column above.

·  If A and B are matrices of the same size then:


·  Suppose that A is an invertible matrix then:


·  A square matrix A is invertible if and only if det(A) ≠ 0. A matrix that is invertible is often called non-singular and a matrix that is not invertible is often called singular.
·  If A is a square matrix then:


·  If A is a square matrix with a row or column of all zeroes then:
det(A) = 0 and so A will be singular.

·  Suppose that A is an n × n triangular matrix then:


·  If two rows (or columns) are interchanged, the sign of the determinant is changed.

Example 1:  For the given matrix below compute both det(A) and det(2A).

Also verify the property det(cA) = cn det(A).
Solution: First of all, we’ll find the scalar multiples of the given matrix.

The determinants:
det(A) = 45
det(2A) = 360 = (8)(45) = 23det(A)
Hence the property is verified.

Example 2:  Let A be an n × n matrix.
(a) det(A) = det(AT)
(b) If two rows (or columns) of A are equal, then det(A) = 0.
(c) If a row (or column) of A consists entirely of 0, then det(A) = 0.
Verify the above properties of determinants for the following matrices:

Solution:

Property (a) holds


Property (b) holds.


Property (c) holds.

Example 3:  Consider the following three matrices.

Verify that det(C) = det(A) + det(B)
Solution: First, notice that we can write C as:


All three matrices differ only in the second row and the second row of C can be found by adding the corresponding entries from the second row of A and B.

The determinants of these matrices are:
det(A) = 15
det(B) = -115
det(C) = -100 = 15 + (-115)
Hence the property is verified.

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