Your school's robotics team mounts three sensors on a robot arm. Sensor C always reports exactly what sensor A and sensor B report added together. Is sensor C telling the robot anything new? Rank answers this kind of question, and determinants are the main tool for finding it.
What Rank Measures
A row is redundant if it is a multiple of another row, or a sum or other combination of other rows. A redundant row still takes up space in the matrix, but it adds no new direction, so it does not count toward rank.
Some problems write rank as $\text{rg}(A)$. It means the same thing as $\operatorname{rank}(A)$.
This matrix has 3 rows, but only 2 of them are independent, so its rank is 2.
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Determine the rank of the following matrix $A$ by observing the relationship between its rows: $$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 1 \end{pmatrix}$$
Determine the rank of the following matrix $A$ by observing the relationship between its rows:
Geometrically, the columns of a $2 \times 2$ matrix are the sides of a parallelogram. The determinant is the (signed) factor by which the matrix scales area.
The unit square (area 1) becomes a parallelogram with area 6, and the determinant is 6. If the two columns point in the same direction, the parallelogram collapses to a segment and the determinant is 0. That is why $\begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} = 4 - 4 = 0$: its rows are proportional, so its rank is only 1.
For $3 \times 3$ matrices there is a diagonal pattern. Some problems call it Sarrus' rule.
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Calculate the value of the determinant of the following matrix $A$ of order 2. This value represents the scale factor of a geometric transformation in the plane: $$A =…
Calculate the value of the determinant of the following matrix $A$ of order 2. This value represents the scale factor of a geometric transformation in the plane:
Given the matrix $M$, calculate the value of the real parameter $x$ so that its determinant is equal to $10$: $$M = \begin{pmatrix} x & 4 \\ 1 & 2 \end{pmatrix}$$
Given the matrix $M$, calculate the value of the real parameter $x$ so that its determinant is equal to $10$:
The factor $(-1)^{i+j}$ gives a checkerboard of signs, starting with $+$ in the top-left corner.
From now on, a minor of order $k$ means the determinant of any $k \times k$ square piece of a matrix. $M_{ij}$ above is the special case of order $n - 1$.
The rules about columns work the same way as the rules about rows.
The scalar rule has a reason: $kA$ multiplies every row by $k$. Each of the $n$ rows contributes one factor of $k$, so you get $k^n$. The product rule is a theorem we use without proof. The inverse rule follows from it: $|A||A^{-1}| = |AA^{-1}| = |I| = 1$.
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If we know that the determinant of a square matrix $A$ of order 2 is $|A| = 5$, what will be the value of the determinant of the matrix $2A$? Apply the properties of…
If we know that the determinant of a square matrix $A$ of order 2 is $|A| = 5$, what will be the value of the determinant of the matrix $2A$? Apply the properties of determinants to justify your answer.
Given a square matrix $A$ of order $3$ whose determinant is $|A| = 4$, use the properties of determinants to calculate with reasoning the value of the following…
Given a square matrix $A$ of order $3$ whose determinant is $|A| = 4$, use the properties of determinants to calculate with reasoning the value of the following determinants:
a) $|A^t \cdot A^{-1}|$
b) $|2 \cdot A \cdot A^t|$
c) The determinant of the matrix resulting from swapping the first two rows of $A$ and multiplying the third column by $-3$.
Given that the value of the determinant of the matrix $3 \times 3$ formed by the column vectors $\vec{c_1}$, $\vec{c_2}$ and $\vec{c_3}$ is equal to $5$, that is,…
Given that the value of the determinant of the matrix $3 \times 3$ formed by the column vectors $\vec{c_1}$, $\vec{c_2}$ and $\vec{c_3}$ is equal to $5$, that is, $|\vec{c_1}, \vec{c_2}, \vec{c_3}| = 5$, calculate without expanding the value of:
For $2 \times 2$: $\operatorname{adj}\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
Combining columns and rank. If $|A| \ne 0$, the columns $C_1, C_2, C_3$ are independent. Any new columns built from them are independent exactly when their coefficient vectors are independent.
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Let $A$ be a square matrix of order $n=4$ such that its determinant is $|A| = \frac{1}{2}$. Using exclusively the properties of determinants, calculate the value of the…
Let $A$ be a square matrix of order $n=4$ such that its determinant is $|A| = \frac{1}{2}$. Using exclusively the properties of determinants, calculate the value of the determinant of matrix $B$, defined as:
$$B = (2 \cdot A^t \cdot Adj(A))^{-1}$$
where $A^t$ is the transpose of $A$ and $Adj(A)$ is its adjugate matrix. Justify each step of your calculation, stating the property used.
Consider the square matrix $A$ of order 3 whose columns are $C_1, C_2, C_3$. It is known that $|A| = 5$. Calculate the value of the determinant of matrix $M$ whose…
Consider the square matrix $A$ of order 3 whose columns are $C_1, C_2, C_3$. It is known that $|A| = 5$. Calculate the value of the determinant of matrix $M$ whose columns are:
If we replace the third column of $M$ with the zero vector, what would be the rank of the new matrix? Justify your answer without performing detailed numerical calculations.
For $4 \times 4$ and bigger, use the three row operations to create zeros in one column, then expand along that column.
Swapping two rows multiplies the determinant by $-1$.
Replacing a row by that row plus a multiple of another row does not change the determinant.
Multiplying a row by $k$ multiplies the determinant by $k$.
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Calculate the value of the following determinant of order $4$ using Gaussian elimination to create zeros in a row or column before expanding by cofactors:…
Calculate the value of the following determinant of order $4$ using Gaussian elimination to create zeros in a row or column before expanding by cofactors:
Calculate, without expanding directly using Sarrus's rule, the value of the following determinant of order 4, applying elementary transformations to make zeros and…
Calculate, without expanding directly using Sarrus's rule, the value of the following determinant of order 4, applying elementary transformations to make zeros and simplify the calculation:
Given the square matrix of order 3: $$A = \begin{pmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ 1. Prove that $|A|$ is independent of…
Given the square matrix of order 3:
$$A = \begin{pmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
Prove that $|A|$ is independent of the value of the variable $x$.
If we define a new matrix $B = A - \lambda I$ (where $I$ is the identity), find the values of $\lambda$ for which the rank of $B$ is less than 3 when $x = \frac{\pi}{4}$.
A robotics engineer is programming the movement of a mechanical arm in three-dimensional space. The sensors detect three force vectors that determine the stability of…
A robotics engineer is programming the movement of a mechanical arm in three-dimensional space. The sensors detect three force vectors that determine the stability of the system. The matrix representing these forces is:
$$M = \begin{pmatrix} k & 1 & 1 & 0 \\ 1 & k & 1 & 2 \\ 1 & 1 & k & k \end{pmatrix}$$
You are asked to:
Determine the rank of matrix $M$ in terms of the real parameter $k$.
Is there any value of $k$ for which the forces are linearly dependent but the rank of the matrix is maximum? Explain your answer based on the theory of minors.
When a matrix has a letter $k$ in it, the rank can change with $k$. Follow this plan:
Find the determinant of the largest square submatrix (or all the largest minors) as an expression in $k$.
Factor it and find its zeros.
For every other value of $k$, the rank is the maximum. For each zero, substitute it in and find the rank with a smaller minor.
For a rectangular matrix, the rank is maximal if any largest minor is nonzero. The rank drops only at values of $k$ that make all of them zero.
What rank 1 and rank 2 look like
Each row of a $3 \times 3$ matrix is a vector in 3D space, drawn as an arrow from the origin. The rank tells you how much room those arrows need.
Rank 1: every row is a scalar multiple of one nonzero vector. All the row vectors are parallel (collinear): they lie along a single line through the origin.
Rank 2: the rows are not all on one line, but they all lie in one plane through the origin (they are coplanar).
Rank 3: the rows are independent and do not fit in any single plane. They point in three truly different directions.
For example, in the first matrix of this page, $R_2 = 2R_1$ so those two arrows are parallel. The third row $(0, 1, 1)$ is not on that line, so together the rows span a plane and the rank is 2.
So when a problem asks for the values of the parameters that make the rank equal to 1, it is asking when all rows become parallel: every $2 \times 2$ minor must be zero, and not every entry is zero.
Some determinants factor nicely without much work. A very useful pattern is the Vandermonde determinant.
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Analyze the rank of the following matrix $A$ in terms of the possible values of the real parameter $k$:…
Analyze the rank of the following matrix $A$ in terms of the possible values of the real parameter $k$:
$A = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$
Determine for which values of $k$ the rank is $1$, $2$, or $3$.
Analyze the linear dependence of the rows of matrix $A$ according to the values of the real parameters $a$ and $b$: $$A = \begin{pmatrix} 1 & a & b \\ a & 1 & b \\ a & b…
Analyze the linear dependence of the rows of matrix $A$ according to the values of the real parameters $a$ and $b$:
$$A = \begin{pmatrix} 1 & a & b \\ a & 1 & b \\ a & b & 1 \end{pmatrix}$$
Determine for which values of $(a, b)$ the rank of $A$ is equal to 1. Describe geometrically what relationship the row vectors have when this occurs.
In a telecommunications system, the signal is lost if the determinant of the transmission matrix becomes zero. The transmission matrix is given by: $$T = \begin{pmatrix}…
In a telecommunications system, the signal is lost if the determinant of the transmission matrix becomes zero. The transmission matrix is given by:
$$T = \begin{pmatrix} 1 & m & m^2 \\ 1 & a & a^2 \\ 1 & b & b^2 \end{pmatrix}$$
Knowing that it is a Vandermonde determinant, express it as a product of linear factors.
If we fix $a=2$ and $b=5$, discuss the rank of $T$ according to the values that the parameter $m$ can take. For what value of $m$ is the rank minimum?
In the design of a graphics engine for a video game, a transformation matrix $M$ is required to be invertible in order to undo movements. Given the matrix:…
In the design of a graphics engine for a video game, a transformation matrix $M$ is required to be invertible in order to undo movements. Given the matrix:
Given the matrices $A$ and $B$ of order $3 \times 3$, it is known that $A$ has rank $3$ and that the determinant of $B$ is $|B| = 0$. a) What is the value of the…
Given the matrices $A$ and $B$ of order $3 \times 3$, it is known that $A$ has rank $3$ and that the determinant of $B$ is $|B| = 0$.
a) What is the value of the determinant of the product $|A \cdot B|$? Justify your answer.
b) Is it possible for the rank of the product matrix $A \cdot B$ to be $3$? Explain why based on the relationship between the determinant and the rank.
Given the matrix: $$A = \begin{pmatrix} \alpha & 1 & 1 \\ 1 & \alpha & 1 \\ 1 & 1 & \alpha \end{pmatrix}$$ 1. Find the values of $\alpha$ for which the matrix does not…
Back to the robot: put each sensor's readings in one row of a matrix. If sensor C always equals A plus B, the matrix is singular and its rank is lower than the number of sensors. Sensor C is redundant, because it adds no independent information.
Summary
Let's practice! 8 medium exercises in this section