- #1
Nusc
- 760
- 2
Prove that, if [tex]AA^T = A^TA = I_n[/tex], then [tex]\det{A} = \pm 1[/tex].
This is daunting.
This is daunting.
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Nusc said:Prove that any scalar multiple of a symmetric matrix is symmetric.
Let [tex]A = (a_i_j)[/tex]
Since [tex]A[/tex] is symmetric, [tex]A = A^T[/tex].
Then [tex]A^T = (a_i_j)[/tex].
Therefore, [tex](cA) = c(A) = c(A^T) = (cA^T)[/tex]
Was it necessary to show that [tex]A = (a_i_j)[/tex] and [tex]A^T = (a_i_j)[/tex]? Is [tex]A^T = (a_i_j)[/tex] even right? I can't express myself mathematically
Nusc said:Prove that, if A and B are two matrices such that A + B and AB are defined, then both A and B are square matrices.
- Let A be an m x r matrix and B an r x n matrix such that,
[tex]A_m_x_rB_r_x_n = (AB)_m_x_n[/tex]
- We know that the sum A + B of the two matrices is the m x n matrix
How do I express them together to show that they are square?
Palindrom said:Since AB is defined, as you yourself wrote, we must have m=n. (Because the product is an mxn*mxn, which is only defined when m=n).
Both matrices are therefor nxn, square matrices!
Nusc said:The r in A is the jth column and in B it's the ith row. So when you say m=n, are you referring to the r's?
And if I were to prove that any scalar multiple of a diagonal matrix is a diagonal matrix, how is that different from, say, letting [tex] A = (a_i_j) [/tex] be any m x n matrix and c any real number?
A diagonal matrix is a square matrix that all of its nonzero entries are on the diagonal.
Then [tex] cA = c(a_i_j) = (ca_i_j)[/tex] but it may not be diagonal.
The basic properties of determinants include linearity, multiplicity, and scaling. Linearity means that the determinant of a sum of matrices is equal to the sum of their determinants. Multiplicity states that if one row or column is multiplied by a constant, the determinant is also multiplied by that constant. Scaling means that if a matrix has a row or column that is a multiple of another row or column, the determinant is equal to zero.
Determinants change in a predictable way when performing elementary row operations. For example, when swapping two rows, the determinant changes sign. When multiplying a row by a constant, the determinant is multiplied by that constant. And when adding a multiple of one row to another row, the determinant remains unchanged.
If the determinant of the coefficient matrix in a system of linear equations is non-zero, then the system has a unique solution. This is known as Cramer's rule. If the determinant is zero, then either the system has no solution or infinitely many solutions.
Yes, determinants can be used to find the area of a parallelogram or the volume of a parallelepiped. By taking the absolute value of the determinant of the coordinates of the points of the shape, the area or volume can be determined.
Yes, determinants have various applications in science including in quantum mechanics, physics, and statistics. In quantum mechanics, determinants are used to describe the state of a quantum system and the probability of different outcomes. In physics, determinants are used to calculate the moment of inertia of a rigid body. In statistics, determinants are used to determine the correlation between variables in a dataset.