# determinant's questions - English 1answer

3.694 determinant questions.

### 1 If $A$ has linearly dependent rows or columns then $\det(A)=0$

1 answers, 42 views linear-algebra matrices determinant

### For which values of the parameter a do the four points (-17, a, 4), (a, 0, 1), (1, -1, 1) and (2,1,0) reside in the same plane. [on hold]

1 answers, 15 views linear-algebra determinant
I've tried several ways of solving this but I can't get the answer correct. the answer is a=-2 and a=-4

### How to prove $\det \begin{bmatrix}A & B \\ C & D \end{bmatrix} =\det(AD-BC)$, where $A, B, C,$ and $D$ are $n\times n$ and commuting. [duplicate]

I can solve for the cases in which $C=0$ or $B=0$. But I cannot find any idea for the case in which both $B$ and $C$ are nonzero matrices.

### How about the $\det(M_B)$, where $M_B(A) = BAB^*$ on $n\times n$ complex matrices.

I know the answer is $$\det M_B = |\det B|^{2n}.$$ I am trying to prove it by considering this operator as a composition of two linear operators right multiplication and left multiplication. Am I ...

### 11 Determinant of a $2\times 2$ block matrix

2 answers, 361 views linear-algebra matrices determinant

### Finding the determinant given the determinant of matrices.

1 answers, 23 views matrices determinant
I am given the determinant of matrix $A$ and matrix $B$. Both are $3 \times 3$ matrices. $\det(A) = -5$ $\det(B) = 5$ I need to find the determinant of $D = 6A^{-1}B^T$. I calculated the ...

### -4 Problems from Linear algebra [on hold]

Let $A=(a_{ij}) \in M_{n}(\mathbb{R})$, $a_{ii} >0$ $~$ for $i$ from $1$ to $n$ and $a_{ij} \leq 0$ for $i\neq j$ ,$1\leq j\leq n$, $\sum_{i=1}^{n}a_{ij} >0$. Prove that $detA>0$

### Finding an orthogonal matrix $P$ of the quadric $3x^2+3z^2-4xy-8xz+4yz=1$

I am trying to find an orthogonal matrix $P$ of the quadric $3x^2+3z^2-4xy-8xz+4yz=1$. I have found that the eigenvalues are $-1$ (with algebraic multiplicity 2) and $8$ (with algebraic multiplicity ...

### 3 How many leading principal minors are there for a $4 \times 4$ matrix?

2 answers, 7.607 views matrices determinant
How many leading principal minors are there for a $4 \times 4$ matrix? Please explain in detail. I know for a $3 \times 3$ matrix.

### 1 How to prove that $\det \left(\begin{array}{cc} 0 & A\\ -B & I \end{array}\right) = \det(AB)$ [closed]

2 answers, 57 views linear-algebra determinant
Show that $$\det \left(\begin{array}{cc} 0 & A\\ -B & I \end{array}\right) = \det(AB)$$ where A, B are compatible matrices, 0 and I are zero and identity matrices of the appropriate size. ...

### 1 Equality of determinants of two square matrices

1 answers, 48 views matrices determinant

### 1 Find all matrices A such as $|\det(A)| = \prod_{i=1}^n\left( \sum_{i=1}^n |a_{i,j}| \right)$

1 answers, 55 views linear-algebra matrices determinant