Read the following paragraph carefully and answer the following questions. $$ \mathrm{A}=\left|\begin{array}{lll} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{array}\right| $$ if, \(\mathrm{U}_{1}, \mathrm{U}_{2}\) and \(\mathrm{U}_{3}\) are column matrices satisfying $$ \mathrm{AU}_{1}=\left|\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right|, \mathrm{AU}_{2}=\left|\begin{array}{l} 2 \\ 3 \\ 0 \end{array}\right|, \mathrm{AU}_{3}=\left|\begin{array}{l} 2 \\ 3 \\ 1 \end{array}\right| $$ and \(\mathrm{U}\) is \(3 \times 3\) matrix whose columns are \(\mathrm{U}_{1}, \mathrm{U}_{2}, \mathrm{U}_{3}\), then The value of \(|\mathrm{U}|\) is \(\ldots \ldots\) (a) 3 (b) \(-3\) (c) \((3 / 2)\) (d) 2

Step by step solution

01

Review Matrix Multiplication Concepts

First, recall how matrix multiplication works: The element at the ith row and jth column of the resulting matrix is calculated by multiplying elements of the ith row of the first matrix with the corresponding elements of the jth column of the second matrix, and then summing them up.

02

Calculate the Column Matrices

We will use our matrix A and the result matrices to calculate the column matrix U. For example, to find \(U_1\), we only have to solve the equation \(Au_{1} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\). Since A is a lower triangular matrix with all ones on the diagonal, a column vector that when multiplied with A gives the first unit vector is itself the first unit vector. Therefore, while multiplying \(U_1\) with A, you get that \(U_1\) equals \(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\). Next, repeat the same steps using the other two equations to find that \(U_2\) equals \(\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\) and \(U_3\) equals \(\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\).

03

Construct the U Matrix

We can now form matrix U by placing matrices \(U_1, U_2, U_3\) column by column. Therefore, matrix U will be: \[ U = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

04

Finding the Determinant of Matrix U

The determinant of U is calculated by observing that matrix U is the identity matrix, which is a special square matrix with ones on the main diagonal and zeros elsewhere. The determinant of the identity matrix is always 1. Thus, the correct answer is none of the (a), (b), (c), or (d) options because none of them are 1.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!