If | x^2+x x+1 x-2 2 x^2+3 x-1 3 x 3 x-3 x^2+2 x+3 2 x-1

step 1 This is given determinant is given as equal to a x plus b from the comparison here we have to fi

If | x^2+x x+1 x-2 2 x^2+3 x-1 3 x ...

If $\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=A x+B$, then (A) $A=\left|\begin{array}{lll}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & 2\end{array}\right|$ (B) $B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & -1\end{array}\right|$ (C) $A=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & 2\end{array}\right|$ (D) $B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & -1\end{array}\right|$

If $\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=A x+B$, then
(A) $A=\left|\begin{array}{lll}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & 2\end{array}\right|$
(B) $B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & -1\end{array}\right|$
(C) $A=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & 2\end{array}\right|$
(D) $B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & -1\end{array}\right|$

Key Concepts

Polynomial Representation of Determinants

When the entries of a matrix depend on a variable (for instance, as polynomials), the overall determinant becomes a polynomial function of that variable. The coefficients of this polynomial (such as those corresponding to the linear term and the constant term) can be determined by algebraic manipulation, term extraction, or by choosing specific values for the variable.

Elementary Row Operations in Determinant Evaluation

Elementary row operations, such as adding multiples of one row to another or factoring out common terms, can simplify a matrix before computing its determinant. While some operations change the value of the determinant in a predictable way, others (like adding a multiple of one row to another) do not alter it, making these techniques very useful in isolating specific terms or coefficients in a polynomial determinant.

Matrix Determinant

The determinant is a scalar value computed from a square matrix that encapsulates important information about the matrix, such as whether it is invertible and how it scales volume when viewed as a linear transformation. It is calculated using a combination of the matrix elements according to specific algebraic rules, often involving permutation sums or cofactor expansion.

Linearity of Determinants

A key property of determinants is their multilinearity, meaning that the determinant is a linear function with respect to the entries of any given row (or column) when the other rows (or columns) are fixed. This property allows one to extract coefficients when the entries are polynomials in a variable, simplifying the analysis of how the determinant changes with that variable.

Transcript

Please give Ace some feedback