If x1 ≠0, x2 ≠0, x3 ≠0, then the determinant

step 1 All right, for we have x1 not equal to 0, x2 not equal to 0, x2 not equal to 0. We'll evaluate t

If $x_{1} \neq 0, x_{2} \neq 0, x_{3} \neq 0$, then the determinant $\left|\begin{array}{ccc}x_{1}+a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} \\ a_{2} b_{1} & x_{2}+a_{2} b_{2} & a_{2} b_{3} \\ a_{3} b_{1} & a_{3} b_{2} & x_{3}+a_{3} b_{3}\end{array}\right|$ is equal to (A) $x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)$ (B) $-x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)$ (C) $x_{1} x_{2} x_{3}\left(1-\frac{a_{1} b_{1}}{x_{1}}-\frac{a_{2} b_{2}}{x_{2}}-\frac{a_{3} b_{3}}{x_{3}}\right)$ (D) None of these

If $x_{1} \neq 0, x_{2} \neq 0, x_{3} \neq 0$, then the determinant
$\left|\begin{array}{ccc}x_{1}+a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} \\ a_{2} b_{1} & x_{2}+a_{2} b_{2} & a_{2} b_{3} \\ a_{3} b_{1} & a_{3} b_{2} & x_{3}+a_{3} b_{3}\end{array}\right|$ is equal to
(A) $x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)$
(B) $-x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)$
(C) $x_{1} x_{2} x_{3}\left(1-\frac{a_{1} b_{1}}{x_{1}}-\frac{a_{2} b_{2}}{x_{2}}-\frac{a_{3} b_{3}}{x_{3}}\right)$
(D) None of these

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Key Concepts

Matrix Determinant Lemma

This theorem provides a formula for the determinant of a matrix modified by a rank-one update. Specifically, if an invertible matrix is perturbed by adding the outer product of two vectors, the determinant of the new matrix can be easily computed using the original determinant and an adjustment factor derived from the vectors and the inverse of the original matrix. This lemma is particularly useful in problems that involve a diagonal matrix plus a rank-one update.

Rank-One Update

A rank-one update refers to the addition of a matrix that has rank one, often expressed as the product of a column vector and a row vector, to another matrix. This concept simplifies calculations involving determinants and inverses because the low rank of the perturbation allows for specialized formulas like the matrix determinant lemma to be applied effectively.

Diagonally Dominant or Diagonal Matrices

Understanding the properties of diagonal matrices is crucial because their determinants are the product of their diagonal entries, making them straightforward to compute. When a diagonal matrix is altered by a rank-one update, the simplicity of the diagonal structure aids in applying the matrix determinant lemma, since the inverse of a diagonal matrix is also diagonal and simple to handle.

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