# Determinants and inverses: Application¶

## Labour market transitions¶

Recall, the discussion (Section 5.1) on using matrices to describe the dynamic evlution of the labour market, and Krusell, Mukoyama, Rogerson and Sahin (2011, JET) estimates for average monthly transition probabilities in the US summarized below

\begin{align}E_{t+1}&=0.962 E_t+0.276 U_t+0.044 N_t\\&\\U_{t+1}&=0.013 E_t+0.501 U_t+0.027 N_t\\&\\N_{t+1}&=0.025 E_t+0.223 U_t+0.929 N_t\end{align}\quad\Rightarrow\quad \begin{pmatrix}E_{t+1}\\ \\U_{t+1}\\ \\N_{t+1}\end{pmatrix}=\begin{pmatrix}0.962&0.276&0.044\\ & &\\0.013&0.501&0.027\\& &\\ 0.025&0.223&0.929\end{pmatrix}\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}

# Determinants and inverses: Application¶

## Labour market transitions¶

Let again

$$\mathbf{A}\equiv\begin{pmatrix}0.962&0.276&0.044\\0.013&0.501&0.027\\ 0.025&0.223&0.929\end{pmatrix}$$

be the matrix of transition probabilities.

For a number of reasons (e.g. see the discussion in Section 5.3) we might be interested in finding the determinant and inverse of $\mathbf{A}$

# Determinant of a 3x3 matrix¶

Given

$$\mathbf{B}\equiv\begin{pmatrix}a&b&c\\d&e&f\\ g&h&i\end{pmatrix}$$

the determinant of $\mathbf{B}$, denoted $|\mathbf{B}|$, can be obtained as

$$|\mathbf{B}| = a(ei-fh)-b(di-fg)+c(dh-eg)$$

E.g. Given

$$\mathbf{A}\equiv\begin{pmatrix}0.962&0.276&0.044\\0.013&0.501&0.027\\ 0.025&0.223&0.929\end{pmatrix}\Rightarrow |\mathbf{A}|=0.43838$$

# Inverse of a 3x3 matrix¶

Step 1: Given

$$\mathbf{B}\equiv\begin{pmatrix}a&b&c\\d&e&f\\ g&h&i\end{pmatrix}$$

the minor matrix, $\mathbf{M_B}$, associated with $\mathbf{B}$ is defined as

$$\mathbf{M_B}\equiv\begin{pmatrix}\left|\begin{matrix}e&f\\h&i\end{matrix}\right|&\left|\begin{matrix}d&f\\g&i\end{matrix}\right|&\left|\begin{matrix}d&e\\g&h\end{matrix}\right|\\\left|\begin{matrix}b&c\\h&i\end{matrix}\right|&\left|\begin{matrix}a&c\\g&i\end{matrix}\right|&\left|\begin{matrix}a&b\\g&h\end{matrix}\right|\\ \left|\begin{matrix}b&c\\e&f\end{matrix}\right|&\left|\begin{matrix}a&c\\d&f\end{matrix}\right|&\left|\begin{matrix}a&b\\d&e\end{matrix}\right|\end{pmatrix}$$

# Inverse of a 3x3 matrix¶

Step 2: Given

$$\mathbf{B}\equiv\begin{pmatrix}a&b&c\\d&e&f\\ g&h&i\end{pmatrix}$$

the cofactor matrix, $\mathbf{C_B}$, associated with $\mathbf{B}$ is defined as

$$\mathbf{C_B}\equiv\begin{pmatrix}\left|\begin{matrix}e&f\\h&i\end{matrix}\right|&-\left|\begin{matrix}d&f\\g&i\end{matrix}\right|&\left|\begin{matrix}d&e\\g&h\end{matrix}\right|\\-\left|\begin{matrix}b&c\\h&i\end{matrix}\right|&\left|\begin{matrix}a&c\\g&i\end{matrix}\right|&-\left|\begin{matrix}a&b\\g&h\end{matrix}\right|\\ \left|\begin{matrix}b&c\\e&f\end{matrix}\right|&-\left|\begin{matrix}a&c\\d&f\end{matrix}\right|&\left|\begin{matrix}a&b\\d&e\end{matrix}\right|\end{pmatrix}$$

# Inverse of a 3x3 matrix¶

Step 3: Given

$$\mathbf{B}\equiv\begin{pmatrix}a&b&c\\d&e&f\\ g&h&i\end{pmatrix}$$

the inverse matrix of $\mathbf{B}$, $\mathbf{B^{-1}}$, can be calculated as

$$\mathbf{B^{-1}}=\frac{1}{|\mathbf{B}|}\mathbf{C_B}^T$$
In [157]:
latexify("A=$A")  Out[157]:$A = \left[ \begin{array}{ccc} 0.962 & 0.276 & 0.044 \\ 0.013 & 0.501 & 0.027 \\ 0.025 & 0.223 & 0.929 \\ \end{array} \right]$In [156]: latexify("A^-1=(1/$detA)*$invA_t_det_t^T=(1/$detA)*$invA_t_det=$invA")

Out[156]:
$A^{-1} = \frac{1}{0.438} \cdot \left[ \begin{array}{ccc} 0.459 & -0.011 & -0.01 \\ -0.247 & 0.892 & -0.208 \\ -0.014 & -0.025 & 0.478 \\ \end{array} \right]^{T} = \frac{1}{0.438} \cdot \left[ \begin{array}{ccc} 0.459 & -0.247 & -0.014 \\ -0.011 & 0.892 & -0.025 \\ -0.01 & -0.208 & 0.478 \\ \end{array} \right] = \left[ \begin{array}{ccc} 1.048 & -0.563 & -0.033 \\ -0.026 & 2.036 & -0.058 \\ -0.022 & -0.474 & 1.091 \\ \end{array} \right]$