Rank of a Matrix
At a monitoring station on Lake Enriquillo, a technician installs several sensors. Each one sends a row of measurements, and the technician suspects that one of them just repeats what the others already say. How can they tell how many sensors provide truly new information? That question is answered by the rank of a matrix, and computing it requires determinants. We start by reviewing them.
Determinants of Order 2 and 3
The determinant only exists for square matrices. For order 2, you multiply the diagonals and subtract:
$$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$For example, $\begin{vmatrix} 6 & -1 \\ 2 & 5 \end{vmatrix} = 6\times 5 - (-1)\times 2 = 32$.
For order 3 we use Sarrus' rule: add the three products along the direction of the main diagonal and subtract the three along the opposite direction.
$$\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}$$Calculate the determinant of the following matrix $A$ of order 2. This value represents the scale factor of a geometric transformation in the plane: $$A =…
Calculate the determinant of the following matrix $A$ of order 2. This value represents the scale factor of a geometric transformation in the plane:
$$A = \begin{pmatrix} 5 & -2 \\ 3 & 4 \end{pmatrix}$$Given the matrix $M$, calculate the value of the real parameter $x$ so that its determinant is equal to $10$: $$M = \begin{pmatrix} x & 4 \\ 1 & 2 \end{pmatrix}$$
Given the matrix $M$, calculate the value of the real parameter $x$ so that its determinant is equal to $10$:
$$M = \begin{pmatrix} x & 4 \\ 1 & 2 \end{pmatrix}$$Find the determinant of the matrix $B$ using Sarrus' rule: $$B = \begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 0 & 4 & 2 \end{pmatrix}$$
Find the determinant of the matrix $B$ using Sarrus' rule:
$$B = \begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 0 & 4 & 2 \end{pmatrix}$$Cofactors, Expansion Along a Row, and Triangular Matrices
The minor $M_{ij}$ of an entry $a_{ij}$ is the determinant that remains after crossing out its row $i$ and its column $j$. The cofactor adds a sign:
$$A_{ij} = (-1)^{i+j} M_{ij}$$The signs form a checkerboard: $+$ in the upper left corner, then alternating.
$$\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$$In the matrix $C$, identify the element $a_{23}$ and calculate its cofactor $A_{23}$: $$C = \begin{pmatrix} 2 & 1 & 5 \\ 0 & -1 & 3 \\ 4 & 2 & 1 \end{pmatrix}$$
In the matrix $C$, identify the element $a_{23}$ and calculate its cofactor $A_{23}$:
$$C = \begin{pmatrix} 2 & 1 & 5 \\ 0 & -1 & 3 \\ 4 & 2 & 1 \end{pmatrix}$$Quickly calculate the determinant of the following upper triangular matrix, giving reasons for your answer: $$T = \begin{pmatrix} 3 & 7 & 9 \\ 0 & -2 & 1 \\ 0 & 0 & 5…
Quickly calculate the determinant of the following upper triangular matrix, giving reasons for your answer:
$$T = \begin{pmatrix} 3 & 7 & 9 \\ 0 & -2 & 1 \\ 0 & 0 & 5 \end{pmatrix}$$Solve the following quadratic equation expressed in determinant form: $\begin{vmatrix} x & 1 & 3 \\ 1 & x & 2 \\ 3 & 2 & 1 \end{vmatrix} = 0$ Find the exact solutions…
Solve the following quadratic equation expressed in determinant form:
$\begin{vmatrix} x & 1 & 3 \\ 1 & x & 2 \\ 3 & 2 & 1 \end{vmatrix} = 0$
Find the exact solutions (they may contain roots).
Properties of Determinants
These properties let you compute determinants without expanding them. Let $A$ be a square matrix of order $n$.
| Operation | Effect on the determinant |
|---|---|
| Transposing | $\|A^t\|=\|A\|$ |
| Swapping two rows (or columns) | Changes sign |
| Multiplying one row (or column) by $k$ | Multiplied by $k$ |
| Multiplying the whole matrix by $k$ | $\|kA\|=k^n\,\|A\|$ |
| Adding a multiple of one row to another | Does not change |
| Two equal or proportional rows, or a row of zeros | The determinant is $0$ |
| Product of matrices | $\|AB\|=\|A\|\,\|B\|$ |
| Inverse (if it exists) | $\|A^{-1}\|=\dfrac{1}{\|A\|}$ |
Given that the determinant of a square matrix $A$ of order 2 is $|A| = 5$, what is the value of the determinant of the matrix $2A$? Apply the properties of determinants…
Given that the determinant of a square matrix $A$ of order 2 is $|A| = 5$, what is 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) $|A^t \times A^{-1}|$
- b) $|2 \times A \times A^t|$
- c) The determinant of the matrix resulting from swapping the first two rows of $A$ and multiplying the third column by $-3$.
Knowing 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,…
Knowing 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:
$|\vec{c_1} + \vec{c_3}, \quad 2\vec{c_2}, \quad \vec{c_1} - \vec{c_3}|$
Justify each step mentioning the property of determinants that you have applied.
Properties with Columns, the Adjugate, and Special Matrices
There are three more ideas that show up in reasoning problems.
Linearity in columns. If a column is a sum, the determinant splits into two:
$$|C_1 + C_1',\ C_2,\ C_3| = |C_1, C_2, C_3| + |C_1', C_2, C_3|$$If after splitting some column is repeated or proportional, that determinant is $0$.
Adjugate matrix. If $\operatorname{Adj}(A)$ is the matrix of cofactors of $A$ (of order $n$), then $A\cdot\operatorname{Adj}(A)^t=|A|\,I$. Taking determinants:
$$|\operatorname{Adj}(A)|=|A|^{\,n-1}$$Vandermonde determinant.
$$\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(b-a)(c-a)(c-b)$$It is zero exactly when two of the numbers $a, b, c$ are equal.
Let $A$ be a square matrix of order $n=4$ such that its determinant is $|A| = \frac{1}{2}$. Using only 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 only the properties of determinants, calculate the value of the determinant of the matrix $B$, defined as:
$$B = (2 \times A^t \times Adj(A))^{-1}$$where $A^t$ is the transpose of $A$ and $Adj(A)$ is its adjugate matrix. Justify each step of your calculation, indicating the property applied.
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 the 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 the matrix $M$ whose columns are:
$$M = (C_1 + C_2, \quad C_2 + C_3, \quad C_3 + C_1)$$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.
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. Show that $|A|$ is independent of…
Given the square matrix of order 3:
$$A = \begin{pmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{pmatrix}$$-
Show that $|A|$ is independent of the value of the variable $x$.
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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 $x = \frac{\pi}{4}$.
Determinants of Order 4: Creating Zeros
For order 4 there is no Sarrus. The practical method is Gauss' method: using the fact that adding a multiple of one row to another does not change the determinant, you create zeros in a column and expand along that column.
This same method works with letters. If you subtract one row from the others and zeros appear everywhere, the determinant factors out effortlessly.
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:
$|D| = \begin{vmatrix} 1 & 2 & 0 & 3 \\ -1 & 3 & 1 & 4 \\ 2 & 0 & -2 & 1 \\ 0 & 1 & 3 & 2 \end{vmatrix}$
Calculate, without expanding directly by Sarrus' rule, the value of the following determinant of order 4, applying elementary transformations to create zeros and…
Calculate, without expanding directly by Sarrus' rule, the value of the following determinant of order 4, applying elementary transformations to create zeros and simplify the calculation:
$$\Delta = \begin{vmatrix} 1 & 1 & 1 & 1 \\ 1 & 1+a & 1 & 1 \\ 1 & 1 & 1+b & 1 \\ 1 & 1 & 1 & 1+c \end{vmatrix}$$What condition must $a, b$ and $c$ satisfy for the rank of the associated matrix to be exactly 2?
In a telecommunications system, the signal is lost if the determinant of the transmission matrix vanishes. The transmission matrix is given by: $$T = \begin{pmatrix} 1 &…
In a telecommunications system, the signal is lost if the determinant of the transmission matrix vanishes. The transmission matrix is given by:
$$T = \begin{pmatrix} 1 & m & m^2 \\ 1 & a & a^2 \\ 1 & b & b^2 \end{pmatrix}$$- a) Knowing that $|T|$ is a Vandermonde determinant, express it as a product of linear factors.
- b) If we set $a=2$ and $b=5$, discuss the rank of $T$ according to the values the parameter $m$ can take. For which values of $m$ is the rank minimal, and what is that minimum rank?
What the Rank of a Matrix Measures
If one row is a multiple of another, or the sum of two others, it does not contribute a new direction: it is "extra" as far as rank is concerned, even though it takes up a row of the matrix.
Determine the rank of the following matrix $A$ by observing the relationship between its rows: $$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 1 \end{pmatrix}$$
Determine the rank of the following matrix $A$ by observing the relationship between its rows:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 1 \end{pmatrix}$$Calculate the rank of the matrix $B$ by finding the highest-order minor that is nonzero: $$B = \begin{pmatrix} 1 & 0 & 5 \\ -2 & 1 & 3 \end{pmatrix}$$ a) State the…
Calculate the rank of the matrix $B$ by finding the highest-order minor that is nonzero:
$$B = \begin{pmatrix} 1 & 0 & 5 \\ -2 & 1 & 3 \end{pmatrix}$$- a) State the highest possible order of a minor of $B$ and how many minors of that order can be formed.
- b) Calculate one of those minors and check whether it is nonzero.
- c) Write the rank of $B$ and justify your answer.
Rank Using Minors
Determine the rank of the matrix $B$ according to the values of the real parameter $a$:…
Determine the rank of the matrix $B$ according to the values of the real parameter $a$:
$B = \begin{pmatrix} 1 & a & 1 & 2 \\ 2 & -1 & a & 1 \\ 1 & 10 & -3 & 5 \end{pmatrix}$
Remember that for a matrix of dimensions $3 \times 4$, the maximum possible rank is $3$.
The Bordered Minors Method
Trying all the minors takes very long. Instead, you start from a nonzero minor and border it: you add one row and one column, and compute the new determinant.
A robotics engineer is programming the movement of a mechanical arm in three-dimensional space. The sensors detect three force vectors that determine the stability of…
A robotics engineer is programming the movement of a mechanical arm in three-dimensional space. The sensors detect three force vectors that determine the stability of the system. The matrix representing these forces is:
$$M = \begin{pmatrix} k & 1 & 1 & 0 \\ 1 & k & 1 & 2 \\ 1 & 1 & k & k \end{pmatrix}$$You are asked to:
-
Determine the rank of the matrix $M$ in terms of the real parameter $k$.
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Is there any value of $k$ for which the forces are linearly dependent but the rank of the matrix is maximum? Explain your answer based on the theory of minors.
Rank According to a Parameter
When the matrix depends on a letter $k$, the rank can change depending on its value. The strategy is always the same: compute the largest possible determinant, factor it, and study what happens at its roots.
Determine the rank of the following matrix $A$ depending on the possible values of the real parameter $k$:…
Determine the rank of the following matrix $A$ depending on the possible values of the real parameter $k$:
$A = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$
Determine for which values of $k$ the rank is $1$, $2$ or $3$.
Study the linear dependence of the rows of the matrix $A$ according to the values of the real parameters $a$ and $b$: $$A = \begin{pmatrix} 1 & a & b \\ a & 1 & b \\ a &…
Study the linear dependence of the rows of the 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.
Rank, Determinant, and Inverse
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:
$M = \begin{pmatrix} 1 & 0 & -1 \\ 0 & m & 3 \\ 4 & 1 & -m \end{pmatrix}$
- a) Find the values of the real parameter $m$ for which the matrix $M$ does not possess an inverse.
- b) For the value $m = 2$, calculate the value of the determinant of the inverse matrix, $|M^{-1}|$.
Given the $3 \times 3$ matrices $A$ and $B$, you know that $A$ has rank $3$ and that the determinant of $B$ is $|B| = 0$. a) What is the value of the determinant of the…
Given the $3 \times 3$ matrices $A$ and $B$, you know 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 $|A \times B|$? Justify your answer.
- b) Is it possible for the rank of the product matrix $A \times B$ 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…
Given the matrix:
$$A = \begin{pmatrix} \alpha & 1 & 1 \\ 1 & \alpha & 1 \\ 1 & 1 & \alpha \end{pmatrix}$$-
Find the values of $\alpha$ for which the matrix does not have an inverse.
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For the values found in the previous section, calculate the rank of the matrix $A^2$.
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Is it possible to find a value of $\alpha$ such that $rg(A) \neq rg(A^2)$? Justify your answer based on the properties of rank and determinants.
Back to the beginning: if you place the sensors' measurements in a matrix (one row per sensor), the rank tells you how many provide independent information. The ones that are left over are linear combinations of the others.