91Ó°ÊÓ

Determine the ranks of: a. \(\left[\begin{array}{rrrr}7 & 9 & -8 & 4 \\ 0 & 1 & 5 & 9 \\ 0 & 0 & 3 & 7 \\ 0 & 0 & 0 & -2\end{array}\right]\) b. \(\left[\begin{array}{rrrr}7 & 9 & -8 & 4 \\ 0 & 0 & 5 & 9 \\ 0 & 0 & 3 & 7 \\ 0 & 0 & 0 & -2\end{array}\right]\) c. \(\left[\begin{array}{rrrr}7 & 9 & -8 & 4 \\ 0 & 0 & 5 & 9 \\ 0 & 0 & 3 & 7 \\ 0 & 0 & 0 & 0\end{array}\right]\) d. Determine the rank of an arbitrary \(4 \times 4\) matrix of the form $$ \left[\begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ 0 & a_{22} & a_{23} & a_{24} \\ 0 & 0 & a_{33} & a_{34} \\ 0 & 0 & 0 & a_{44} \end{array}\right] $$ in terms of the entries \(a_{11}, a_{22}, a_{33}\), and \(a_{44}\). e. Generalize the result of part d to \(n \times n\) matrices in which the only nonzero entries are in the upper right triangular region of the matrix.

Step by step solution

01

Find the rank for matrix a

Examine the main diagonal of the matrix. Since none of the elements is zero, the rank of matrix a is 4.

02

Find the rank for matrix b

Inspect the diagonal elements of this matrix. Since no element is zero, the rank of the matrix b is also 4.

03

Find the rank for matrix c

In the case of matrix c, we have a zero in the main diagonal. So, when counting the number of non-zero elements on the diagonal, we find the rank of the matrix c to be 3.

04

Determine the rank of an arbitrary 4x4 upper triangular matrix

For an arbitrary 4x4 triangle matrix, the rank will be equal to the number of non-zero elements along the main diagonal. So, the matrix rank would be 4 if and only if none of \(a_{11},a_{22},a_{33},a_{44}\) is a zero.

05

Generalize for n x n upper triangular matrices

For a generalized case of an n x n upper triangular matrix, the number of non-zero elements on the main diagonal will determine the rank of the matrix. So if there are k non-zero elements on the main diagonal, the rank will be k.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix. In this matrix, all elements below the main diagonal are zero. This means if you have a matrix represented by \(\begin{array}{cccc} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \ 0 & a_{22} & a_{23} & \cdots & a_{2n} \ 0 & 0 & a_{33} & \cdots & a_{3n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \cdots & a_{nn} \end{array} \)

This outer structure simplifies many computations. For example, when determining the rank of such a matrix, you focus mainly on the entries along the main diagonal. Upper triangular matrices are widely useful in linear algebra, particularly in solving systems of linear equations. With these matrices, back-substitution is a straightforward process, making calculations more efficient.

Diagonal Elements

The diagonal elements of a matrix are those that lie on the path from the upper-left to the lower-right of a square matrix. In terms of indices in a matrix, these are the elements where the row number equals the column number, often written as \(a_{ii}\).

For an upper triangular matrix, these elements are crucial because they determine the matrix's rank. If you're inspecting a matrix like:\(\begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \ 0 & a_{22} & a_{23} & a_{24} \ 0 & 0 & a_{33} & a_{34} \ 0 & 0 & 0 & a_{44} \end{array}\)

The rank relies on how many of these diagonal entries are non-zero. In matrix computations, these diagonal elements serve not only as an indicator of rank but also affect properties like invertibility.

Non-Zero Elements

Non-zero elements in a matrix have a significant impact on determining the matrix's rank. Specifically for upper triangular matrices, where the rank directly correlates with the number of non-zero diagonal elements. This simplifies the calculation process considerably.

For instance, if a diagonal element of an upper triangular matrix is zero, it reduces the rank by 1. This is because the zero element does not contribute to the span of the matrix's rows. For us to find the rank of a matrix, especially an upper triangular one, simply tally the non-zero diagonal elements. This focus minimizes the need to evaluate every element individually, streamlining the process.

n x n Matrix

An \(n \times n\) matrix signifies a square matrix containing \(n\) rows and \(n\) columns. This symmetry allows many unique properties not found in non-square matrices. These matrices lay the groundwork for understanding complex matrix operations and transformations.

In the context of upper triangular matrices, an \(n \times n\) matrix maintains its properties efficiently due to having a systematic zeroing of both sides of the diagonal. With this understanding, generalized computations such as determining rank become straightforward. Exploiting the properties of the main diagonal streamlines calculations across any size matrix, simplifying linear algebra tasks down to counting non-zero diagonal elements. Understanding these contributes vastly to algebraic solutions in mathematical and real-world applications.