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Mathematics Question on Determinants

The value of the determinant ∣1+a2−b22ab−2b 2ab1−a2+b22a 2b−2a1−a2−b2∣\begin{vmatrix}1+a^{2}-b^{2}&2ab&-2b\\\ 2ab&1-a^{2}+b^{2}&2a\\\ 2b&-2a&1-a^{2}-b^{2}\end{vmatrix} is equal to

A

0

B

(1+a2+b2)(1+a^2+b^2)

C

(1+a2+b2)2(1+a^2+b^2)^2

D

(1+a2+b2)3(1+a^2+b^2)^3

Answer

(1+a2+b2)3(1+a^2+b^2)^3

Explanation

Solution

. Let Δ=∣1+a2−b22ab−2b 2ab1−a2+b22a 2b−2a1−a2−b2∣\Delta=\begin{vmatrix}1+a^{2}-b^{2} & 2 a b & -2 b \\\ 2 a b & 1-a^{2}+b^{2} & 2 a \\\ 2 b & -2 a & 1-a^{2}-b^{2}\end{vmatrix}
Apply C1→C1−bC3C_{1} \rightarrow C_{1}-b C_{3} and C2C_{2}
→aC3+C2\rightarrow a C_{3}+C_{2}
Δ=∣1+a2−b2+2b22ab−2ab−2b 2ab−2ab1−a2+b2+2a22a 2b−b+a2b+b3−2a+a−a3−ab21−a2−b2∣\Delta=\begin{vmatrix}1+a^{2}-b^{2}+2 b^{2} & 2 a b-2 a b & -2 b \\\ 2 a b-2 a b & 1-a^{2}+b^{2}+2 a^{2} & 2 a \\\ 2 b-b+a^{2} b+b^{3} & -2 a+a-a^{3}-a b^{2} & 1-a^{2}-b^{2}\end{vmatrix}
=∣(1+a2+b2)0−2b 0(1+a2+b2)2a b(1+a2+b2)−a(1+a2+b2)(1−a2−b2)∣=\begin{vmatrix}\left(1+a^{2}+b^{2}\right) & 0 & -2 b \\\ 0 & \left(1+a^{2}+b^{2}\right) & 2 a \\\ b\left(1+a^{2}+b^{2}\right) & -a\left(1+a^{2}+b^{2}\right) & \left(1-a^{2}-b^{2}\right)\end{vmatrix}
=(1+a2+b2)2∣10−2b 012a b−a(1−a2−b2)∣=\left(1+a^{2}+b^{2}\right)^{2}\begin{vmatrix}1 & 0 & -2 b \\\ 0 & 1 & 2 a \\\ b & -a & \left(1-a^{2}-b^{2}\right)\end{vmatrix}
=\left(1+a^{2}+b^{2}\right)^{2}\left\\{\left(1-a^{2}-b^{2}+2 a^{2}\right)+2 b^{2}\right\\}
=(1+a2+b2)2(1+a2+b2)=(1+a2+b2)3=\left(1+a^{2}+b^{2}\right)^{2}\left(1+a^{2}+b^{2}\right)=\left(1+a^{2}+b^{2}\right)^{3}

Determinantshave several characteristics that are highly helpful since they enable us to produce the same outcomes using various and less complex entry (element) combinations. These determinant features make it easier to evaluate by ensuring that there are never more than a certain number of zeros in a row or column. For determinants of any order, these characteristics are true.

The major characteristics of determinants are the following ten: reflection, all-zero, proportionality or repetition, switching, scalar multiple, sum, invariance, factor, triangle, and cofactor matrix features.

A square matrix's decimal value is a determinant. We can assign a real or complex integer to every square matrix. The determinant of the square matrix is this particular integer. It may be viewed as a mapping function that links a certain real or complex integer to a square matrix.

As a result, we can state that a square matrix A with order 'n' and the same number of rows and columns will contain a single real or complex integer that encapsulates significant matrix information. The square matrix A's determinant is represented by this integer. Denoted by det A or |A|.

  • Instead of reading |A| as the absolute value of A, think of it as the determinant A.
  • A matrix does not provide numerical value, although a determinant does.
  • Determinants exclusively belong to Square Matrices since they always have an equal number of rows and columns. A 1 1 matrix's determinant is thus that particular integer.