**A matrix cannot have multiple determinants**since the determinant is a scalar that can be calculated from the elements of a square matrix. Swapping of rows or columns will change the sign of a determinant.

Can a matrix have nullity 0?

**nullity of a matrix**.

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Thus, the value of the determinant of of every matrix is determined by the definition. **There can be only one determinant function**.

If two determinants differ by just one column, we **can add them together** by just adding up these two columns. For example: … All other elementary row operations will not affect the value of the determinant!

Thus, **both the matrices have the same determinant value**. Hence, we cay say, two different matrices can have the same determinant value.

The answer is **“NO”**. The determinant only exists for square matrices.

Since a determinant **stays the same** by interchaning the rows and columns, it should be obvious that similar to ‘row-by-row’ multiplication that we’ve encountered above, we can also have ‘row-by-column’ multiplication and ‘column-by-column’ multiplication.

Therefore, when **we add a multiple of a row to another row**, the determinant of the matrix is unchanged. Note that if a matrix A contains a row which is a multiple of another row, det(A) will equal 0.

You have to split the given determinant into **a sum of two determinants** using the splitting of terms in one row at a time keeping the other rows intact, each of which then gives a sum of two determinants using the splitting of the terms of the next row.

A matrix is said to be singular **if and only if its determinant is equal to zero**. A singular matrix is a matrix that has no inverse such that it has no multiplicative inverse.

Determinants of equivalent **matrices are not same in general** but here you are adding rows ,therefore determinant remains same. Yes, if two matrices are similar then their trace/rank/determinant/eigenvalue will be equal.

Matrices; Determinants and Eigenvalues: Given a square real matrix A its determinant det(A) is the product of the eigenvalues of A . Because of this, **two matrices with the same eigenvalues have the same determinant**.

If a matrix **has the same number of rows and columns** (e.g., if m == n), the matrix is square. The definitions that follow in this section only apply to square matrices.

- Pick any row or column in the matrix. It does not matter which row or which column you use, the answer will be the same for any row. …
- Multiply every element in that row or column by its cofactor and add. The result is the determinant.

Difference between Matrix and Determinant: … A matrix is a group of numbers but **a determinant is** a unique number related to that matrix. In a matrix the number of rows need not be equal to the number of columns whereas, in a determinant, the number of rows should be equal to the number of columns.

**If either two rows or two columns are identical, the determinant equals zero**. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.

The effect of scaling a matrix. Since a linear transformation can always be written as **T(x)=Ax** for some matrix A, applying a linear transformation to a vector x is the same thing as multiplying by a matrix. … In one dimension, the effect of doubling every vector would simply double the expansion of length by ˜T.

**Exchanging two rows, or two columns of a matrix switches** the sign of the determinant. For a fun corollary this means any matrix that has two rows or columns that are the same must have zero determinant.

Matrix multiplication **is not commutative**.

If A is an n × n matrix, and k is any constant, then **detkA = kn detA**. When using the theorem, it is important to keep in mind that the constant k in the determinant formula gets multiplied by itself n times, since each of the n rows of A was multiplied by k.

Proof by induction that **transposing a matrix does not change its determinant**.

If the determinant of a matrix is 0, the matrix is said to be singular, and if the determinant is 1, **the matrix is said to be unimodular**.

determinant: The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 for the unit matrix. Its abbreviation is “det “.

The determinant of the inverse of an invertible matrix is the inverse of the determinant: **det(A-1) = 1 / det(A)** [6.2. 6, page 265]. Similar matrices have the same determinant; that is, if S is invertible and of the same size as A then det(S A S-1) = det(A).

In matrix calculus, Jacobi’s formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. where tr(X) is the trace of the matrix X. It is named after the mathematician Carl Gustav Jacob Jacobi.

In a determinant the sum of the product of the elements of any row (or column) with the cofactors of the corresponding elements of any other row (or column) is **zero**.

We determine whether a matrix is a singular matrix or a non-singular matrix depending on its determinant. The determinant of a matrix ‘A’ is denoted by ‘**det A’ or ‘|A|**‘. If the determinant of a matrix is 0, then it is said to be a singular matrix.

Since A is a singular matrix. So det A = 0. FINAL ANSWER. Hence the required value of a = **4**.

- If the determinant is equal to , the matrix is singular.
- If the determinant is non-zero, the matrix is non-singular.

A matrix’s determinant gives the volume of the parallelepiped whose sides are the columns of the matrix. If two matrices with identical dimensions have the same determinant, it means **their corresponding parallelepipeds have the same volume.**

The matrices can be transformed into one another by a combination of elementary row and column operations. Two matrices are **equivalent if and only if they have the same rank**.

(a) Find a matrix B in reduced row echelon form such that B is row equivalent to the matrix A. … Condition that Two Matrices are Row Equivalent We say that two m×n matrices are row equivalent if one can be obtained from the other by a sequence of elementary **row** operations. Let A and I be 2×2 matrices defined as follows.

Two matrices are equal if all three of the following conditions are met: **Each matrix has the same number of rows**. Each matrix has the same number of columns. Corresponding elements within each matrix are equal.

However, since all non-square matrices do not have a unique inverse, it was not useful to define the determinant for non-square matrices. Only the **square matrices require knowledge of whether they have a unique inverse or not**. So the determinant was only defined for square matrices as a result.

- The definition of a matrix inverse requires commutativity—the multiplication must work the same in either order.
- To be invertible, a matrix must be square, because the identity matrix must be square as well.

Yes, **the determinant of a matrix can be a negative number**. By the definition of determinant, the determinant of a matrix is any real number. Thus, it includes both positive and negative numbers along with fractions.

1.5 So, by calculating the determinant, we get det(A)=ad-cb, Simple enough, now lets take AT (**the transpose**). 1.8 So, det(AT)=ad-cb. 1.9 Well, for this basic example of a 2×2 matrix, it shows that det(A)=det(AT).

Not all 2 × 2 matrices have an inverse matrix. If the determinant of the matrix is zero, then it will not have an inverse; the matrix is then said to be singular. **Only non-singular matrices have inverses**.