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.