Determinants

$$A = \begin{bmatrix} 1 & -7 & 8 & 9 & -6 \\ 0 & 2 & -5 & 7 & 3 \\ 0 & 0 & 2 & 4 & -1 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & -1 & 0 \end{bmatrix}$$ Note: The matrix is not fully upper triangular because $a{43} = 1 \neq 0$. So cofactor expansion is required. --...

$$A = \begin{bmatrix} -2 & 8 & -9 \\ -1 & 7 & 0 \\ 1 & -4 & 2 \end{bmatrix}$$ A determinant is a scalar value associated with every square matrix, obtained from its entries via a well-defined rule. It is denoted $\det(A)$ or $A$. Geometrically it represents...

Matrix: $$A = \begin{bmatrix} 1 & 5 & -6 \\ -1 & -4 & 4 \\ -2 & -7 & 9 \end{bmatrix}$$ Task: define determinant and evaluate $A$ without expanding. A determinant is a scalar-valued function defined on square matrices that assigns to every square matrix a un...

$$A = \begin{bmatrix} 2 & -8 & 6 & 8 \\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 1 & -4 & 0 & 6 \end{bmatrix}$$ Rules used: - Adding a multiple of one row to another does not change the determinant. - Factoring scalar $k$ out of a row means the original determ...

$$A = \begin{bmatrix} 5 & -7 & 2 & 2 \\ 0 & 3 & 0 & -4 \\ -5 & -8 & 0 & 3 \\ 0 & 5 & 0 & -6 \end{bmatrix}$$ Column 3 entries: $a{13}=2,\ a{23}=0,\ a{33}=0,\ a{43}=0$. Only one non-zero term: $$\det(A) = a{13}\cdot(-1)^{1+3}\cdot M{13} = 2\cdot M{13}$$ Delet...