Singular matrix

A matrix is called singular if its determinant is equal to zero. For example, the matrix N=\begin{matrix} 2 & 4 \\ 1 & 2 \end{matrix} is singular, because its determinant is 2\cdot 2 - 4 \cdot 1 = 0.

This is useful when:

finding inverse matrices
finding transformations

Is the matrix \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} singular?

Determinant = ad-bc
a=1, b=2, c=3 and d=4.
Determinant = 1\times 4 - 2\times 3 = 4 - 6 = -2
Answer: No, because the determinant is not zero.

Is the matrix \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} singular?

Determinant = ad-bc
a=1, b=2, c=2 and d=4.
Determinant = 1\times 4 - 2\times 2 = 4 - 4 = 0
Answer: Yes, because the determinant is zero.

Find p for the singular matrix \begin{bmatrix} 4 & p+2 \\ -1 & 3-p \end{bmatrix}

Singular matrices have a determinant of 0.
Determinant = ad-bc
a=4, b=p+2, c=-1 and d=3-p.
0 = 4(3-p) - (p+2)(-1)
0 = 12 - 4p + p + 2
0 = 14 - 3p
3p = 14
p = \frac{14}{3}

Given that \begin{bmatrix} 1 & 2 \\ k & 4 \end{bmatrix} is singular, find k.

Singular matrices have a determinant of 0.
Determinant = ad-bc
a=1, b=2, c=k and d=4.
0 = 1\times 4 - 2\times k
0 = 4 - 2k
2k = 4
k = 2

Find p for the singular matrix \begin{bmatrix} 2 & p-1 \\ 3 & 5 \end{bmatrix}

Singular matrices have a determinant of 0.
Determinant = ad-bc
a=2, b=p-1, c=3 and d=5.
0 = 2\times 5 - (p-1)\times 3
0 = 10 - 3p + 3
0 = 13 - 3p
3p = 13
p = \frac{13}{3}

Inverse matrix

A singular matrix does not have an inverse matrix.

Does the matrix \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix} have an inverse?

A matrix has an inverse if its determinant is not zero.
Determinant = ad-bc
a=1, b=2, c=3 and d=6.
Determinant = 1\times 6 - 2\times 3 = 6 - 6 = 0
Answer: No, because the determinant is zero, so the matrix is singular and doesn’t have an inverse.

Does the matrix \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} have an inverse?

A matrix has an inverse if its determinant is not zero.
Determinant = ad-bc
a=1, b=2, c=3 and d=4.
Determinant = 1\times 4 - 2\times 3 = 4 - 6 = -2
Answer: Yes, because the determinant is not zero, so the matrix is not singular and has an inverse.

Does the matrix \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} have an inverse?

A matrix has an inverse if its determinant is not zero.
Determinant = ad-bc
a=2, b=4, c=1 and d=2.
Determinant = 2\times 2 - 4\times 1 = 4 - 4 = 0
Answer: No, because the determinant is zero, so the matrix is singular and doesn’t have an inverse.

flashcards

Question	Answer
What is a singular matrix?	A matrix is called singular if its determinant is equal to zero.
What is the determinant of matrix \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and is it singular?	Determinant = 1\times 4 - 2\times 3 = 4 - 6 = -2, so it is not singular because the determinant is not zero.
What is the determinant of matrix \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} and is it singular?	Determinant = 1\times 4 - 2\times 2 = 4 - 4 = 0, so it is singular because the determinant is zero.
Find p such that \begin{bmatrix} 4 & p+2 \\ -1 & 3-p \end{bmatrix} is singular.	Set determinant 0 = 4(3-p) - (p+2)(-1) to get 0 = 14 - 3p, so p = \frac{14}{3}.
Given \begin{bmatrix} 1 & 2 \\ k & 4 \end{bmatrix} is singular, find k.	Set 0 = 1\times 4 - 2\times k, so 0 = 4 - 2k, hence k = 2.
Find p for the singular matrix \begin{bmatrix} 2 & p-1 \\ 3 & 5 \end{bmatrix}.	Set 0 = 2\times 5 - (p-1)\times 3 to get 0 = 13 - 3p, so p = \frac{13}{3}.
Does a singular matrix have an inverse?	No, a singular matrix does not have an inverse matrix.
Does matrix \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix} have an inverse?	No, because its determinant is 1\times 6 - 2\times 3 = 0, so it is singular and has no inverse.
Does matrix \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} have an inverse?	Yes, because its determinant is 1\times 4 - 2\times 3 = -2, which is not zero, so it is not singular.
Does matrix \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} have an inverse?	No, because its determinant is 2\times 2 - 4\times 1 = 0, so it is singular and has no inverse.