# What is Determinant properties: Definition and 15 Discussions

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible, and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one).
The determinant of a matrix A is denoted det(A), det A, or |A|.
In the case of a 2 × 2 matrix the determinant can be defined as

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Similarly, for a 3 × 3 matrix A, its determinant is

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{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}
Each determinant of a 2 × 2 matrix in this equation is called a minor of the matrix A. This procedure can be extended to give a recursive definition for the determinant of an n × n matrix, known as Laplace expansion.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped is expressed by a determinant. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.

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1. ### Decompose 4x4 determinant into 24 determinants -- How many are zero?

Here is an example of the decomposition for a 2 x 2 matrix We have ##2^2=4## determinants, each with only #n=2# non-automatically-zero entries. By "non-automatically-zero" I just mean that they aren't zero by default. Of course, any of ##a,b,c##, or ##d## can be zero, but that depends on the...
2. ### Proving ##(cof ~A)^t ~A = (det A)I##

i-th column of ##cof~A## = $$\begin{bmatrix} (-1)^{I+1} det~A_{1i} \\ (-1)^{I+2} det ~A_{2i}\\ \vdots \\ (-1)^{I+n} det ~A_{ni}\\ \end{bmatrix}$$ Therefore, the I-th row of ##(cof~A)^t## = ##\big[ (-1)^{I+1} det~A_{1i}, (-1)^{I+2} det ~A_{2i}, \cdots, (-1)^{I+n} det ~A_{ni} \big]## The I-th...
3. ### I Determinant of a specific, symmetric Toeplitz matrix

Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...
4. ### A What physical meaning can the “determinant” of a divergency have?

I am [working][1] on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\det w=\exp(\operatorname{reg }\ln w)$$ which is analogous to how determinant of a matrix can be...
5. ### Finding the determinant of a matrix using determinant properties

Hi, I have been having some trouble in finding the determinant of matrix A in this Q Which relevant determinant property should I make use of to help me find the determinant of matrix A and maybe matrix B also This is what I have tried for matrix A so far but it's not much help really Any...
6. ### Finding the Determinant to find out if the matrix is invertible

question: My first attempt: my second attempt: So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...
7. ### Does a Positive Definite Hermitian Matrix Always Have a Positive Determinant?

1. Problem statement : suppose we have a Hermitian 3 x 3 Matrix A and X is any non-zero column vector. If X(dagger) A X > 0 then it implies that determinant (A) > 0. I tried to prove this statement and my attempt is attached as an image. Please can anyone guide me in a step by step way to...
8. ### Proof regarding determinant of block matrices

Homework Statement Let A,B,C,D be commuting n-square matrices. Consider the 2n-square block matrix ##M= \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}##. Prove that ##\left | M \right |=\left | A \right |\left | D \right |-\left | B \right |\left | C \right |##. Show that the result may not be...
9. ### Determinant of 3x3 matrix equal to scalar triple product?

The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...
10. ### Having trouble solving using properties of determinants ....

Homework Statement I'm a bit at a loss - I thought the last row with '1's would be useful, but it just gave me: (b2c - bc2) - (a2c - ac2) + (a2b - ab2) and bc(b - c) - ac(a - c) + ab(a - b) But then it is a dead end. I am probably doing something stupid again ... Any help appreciated.
11. ### Matrices and determinant properties?

Homework Statement Homework Equations The Attempt at a Solution I tried to see if the problem has any properties with determinants that i can apply but the properties i learned didn't involve the use of adjoint matrices so I'm kind of stumped on this one. Any hints would be...
12. ### Does Det(AB) = 0 Imply Det(A) or Det(B) Must Be Zero?

1) If det(AB) = 0, is det(A) or det(B) = 0? Give reasons for your answer. Q1) First, cannot both det(A) or det(B) be 0? If it can, is this statement false. In any case, how can I prove that this is true for all statement since I only know how to find an example to show this is true, which...
13. ### [Linear Algebra] Showing equality via determinant properties

Problem: Show, without evaluating directly, that \left|\begin{matrix} a_1+b_1t&a_2+b_2t&a_3+b_3t \\ a_1t+b_1&a_2t+b_2&a_3t+b_3 \\ c_1&c_2&c_3 \end{matrix}\right| = (1-t^2)\left|\begin{matrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \\ c_1&c_2&c_3 \end{matrix}\right| Clearly, here I'm...
14. ### Determinant Equality Explained without Evaluation | Boas 3rd Ed. HW Question 7

Homework Statement Show without evaluating the determinant the equality. Homework Equations \left( \begin{array}{ccc} 1 & a & bc \\ 1 & b & ac \\ 1 & c & ab \end{array} \right) = \left( \begin{array}{ccc}...
15. ### Linear Algebra - Determinant Properties

Homework Statement 1. Give an example of a 2x2 real matrix A such that A^2 = -I 2. Prove that there is no real 3x3 matrix A with A^2 = -I Homework Equations I think these equations would apply here? det(A^x) = (detA)^x det(kA) = (k^n)detA (A being an nxn matrix) det(I) = 1 The...