Is A Invertible When Each Diagonal Element is Nonzero?

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  • Thread starter karush
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In summary, a matrix A is invertible when there exists another matrix A^-1 that, when multiplied by A, results in the identity matrix I. To show that a matrix is invertible, one can calculate the determinant or row reduce it to its RREF. Non-square matrices cannot be invertible. Invertible matrices have unique inverses, the inverse of the inverse is the original matrix, and the inverse of a product of matrices is equal to the product of the inverses in reverse order. It is important to show that a matrix is invertible because it allows for solving systems of linear equations and has many practical applications.
  • #1
karush
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$\textsf{Let }$
$A=\textit{diag} (a_1,a_2,...,a_n)$.
$\textsf{Show that A is invertible iff each}$
$a_i\ne 0.$

$\textsf{Ok I didn't know formally how to answer this.}$
$\textsf{Except i can see that an $a=0$ would mess things up}$
 
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  • #2
I don't know if this helps but a matrix is invertible iff its determinant is non-zero.
 
  • #3
The inverse matrix for the diagonal matrix with [tex]a_1[/tex], [tex]a_2[/tex], … , [tex]a_n[/tex] on the diagonal, is, rather trivially, the diagonal matrix with [tex]1/a_1[/tex], [tex]1/a_2[/tex], …, [tex]1/a_n[/tex] on the diagonal. Show that and you are finished.
 

FAQ: Is A Invertible When Each Diagonal Element is Nonzero?

What does it mean for a matrix to be invertible?

When a matrix A is invertible, it means that there exists another matrix, typically denoted as A^-1, that when multiplied by A results in the identity matrix. In other words, the inverse of A "undoes" the effects of A.

How do you show that a matrix A is invertible?

To show that a matrix A is invertible, you can use several methods such as row reduction, determinants, or the invertibility theorem. For example, if the determinant of A is non-zero, then A is invertible.

What are the benefits of having an invertible matrix?

An invertible matrix has several benefits, including the ability to solve systems of linear equations, find unique solutions to linear transformations, and perform matrix operations such as addition, subtraction, and multiplication.

Can every matrix be inverted?

No, not every matrix can be inverted. In order for a matrix to be invertible, it must be a square matrix (same number of rows and columns) and have a non-zero determinant. If these conditions are not met, the matrix cannot be inverted.

How is the inverse of a matrix calculated?

The inverse of a matrix A is calculated by using the formula A^-1 = (1/det(A)) * adj(A), where det(A) is the determinant of A and adj(A) is the adjugate matrix of A. Alternatively, you can also use row reduction to find the inverse of a matrix.

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