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Cramer’s Rule
Cramer’s rule is a method of solving a system of linear equations through the use of determinants. 
Matrices and Determinants
To use Cramer’s Rule, some elementary knowledge of matrix algebra is required. An array of numbers, such as
		 6 5 a11 a​12
 		A = (
 	 	 3 4 a21 a22
is called a matrix. This is a “2 by 2” matrix. However, a matrix can be of any size, defined by m rows and n columns (thus an “m by n” matrix). A “square matrix,” has the same number of rows as columns. To use Cramer’s rule, the matrix must be square.
A determinant is number, calculated in the following way for a “2 by 2” matrix:
					 
 		a11 a12
 A = 	 	 = a11 a22 - a21 a12	
				a21 a22
					 
For example, letting a​11​ = 6, a​12 = 5, a​21​ = 3, a​22​ = 4: 	
 						 
				 6 5
		 	 (A(= 	= 6 (4) - 3 (5) = 9
				 3 4
					 
For “m by n” matrices of orders larger than 2 by 2, there is a general procedure that can be used to find the determinant. This procedure is best explained as an example. Consider the determinant for a 3 by 3 matrix
			 a11 a12 a13
		(A( = a21 a22 a23 
			 a31 a32 a33				
The determinant (A( is calculated as follows:
				a22 a23	 a31 a23		 a21 a22	
		(A( = a11		 - a12 	 	 + a13 	
				a32 a33	 a31 a33		 a31 a32
						 note the sign change
		(A( = a11 (a22 a33 - a23 a32) - a12 (a21 a33 - a23 a31) + a13 (a21 a32 - a22 a31)
						Sign change (like a “2 by 2” matrix) 
Note: Sign changes alternate, following the order: positive, negative, positive, negative, etc.
The determinant of the 3 by 3 matrix is the sum of three products. The first step is to understand the placement of the elements from the matrix into the determinant equation. This is done by: 
The three products to be summed correspond to the three elements along the top 
 row of the matrix (this would be a11, a12, a13).
2. Now, imagine a line that goes though the top row of elements (see the model 
	 below).
Beginning at a11, imagine, too, a line through the first column (Figure 1).
The 4 remaining elements are used to construct a new “2 by 2” matrix, and the element a11 is used to form the first of the three parts of the calculation:
				 a22 a23	
			 a11 		
				 a32 a33	
	5. The same process (follow steps 1-4 above) is then repeated for a12 and a13 as seen 
 in figures 2 and 3 respectively, i.e., the top row contains the element used to 
multiply the new “2 by 2” matrix, and the column which contains the element 
	 from the top row is omitted.
		a11 a12 a13		a11 a12 a13		a11 a21 a31
		a21 a22 a23		a21 a22 a23		a21 a22 a23
		a31 a32 a33		a31 a32 a33		a31 a32 a33
 	 	 Figure 1		 Figure 2		 Figure 3
For an example, consider:
				5 6 7
			A = 	2 1 4
				9 6 3
Find the determinant (A(.
Determinant (A( is calculated as follows:
				 1 4		 2 4		 2 1
		 (A( = 5	 	 - 6 	 + 7 		
				 6 3		 9 3		 9 6
( A( = 5 [1 (3) - 6 (4)] - 6 [2 (3) - 9 (4) ] + 7 [2 (6) - 9 (1)]		
			
 		 (A( = 96
A Description of Cramer’s Rule 
Cramer’s rule is a method of solving a system of linear equations through the use of determinants. Cramer’s rule is given by the equation
		xi = (Ai(	
			(A(
where xi is the i th endogenous variable in a system of equations, (A( is the determinant of the original A matrix as discussed in the previous section, and (Ai( is the determinant a special matrix formed as part of Cramer’s rule.
To use Cramer’s rule, two (or more) linear equations are arranged in the matrix form 
A x = d. For a two equation model:
		 A	 x	= d	
		a11 a12 x1 	 d1	
		a21 a22 x2 = d2	
A is the matrix corresponding to the number of equations in a system (here, two equations), and the number of endogenous variables in the system (here 2 variables). Remember that the matrix must be square, so the number of equations must equal the same number of endogenous variables. Position x has one column and corresponds to the number of endogenous variables in the system. Finally, position d contains the exogenous terms of each linear equation.
Note: The determinant for a matrix must not equal 0 ((A( ( 0). If (A( = 0 then there is no solution, or there are infinite solutions (from dividing by zero). Therefore, (A( ( 0. When A ( 0, then a unique solution exists.
Applying Cramer’s Rule in a 2x2 example
	
Using Cramer’s rule to solve for the unknowns in the following linear equations:
		2x1 + 6x2 = 22
		
		-x1 + 5x2 = 53
	Then,
			A x = d
			
			2 6 x1	 22	
				 = 	
			-1 5 x2 53
					 2 6	
	The primary determinant (A( = 	 = 2 (5) - (-1) 6 = 16	
					 -1 5	
We need to construct xi =	(Ai(, for i=1 and for i=2.
 		 	(A(
	 	
The first special determinant A1 is found by replacing the first column of the primary matrix with the constant ‘d’ column. The new special matrix A1 now appears as:
		
			22 6
A1 = 
			53 5
and solved as a regular matrix determinant,
		(A1( = 22 (5) - 53 (6) = -208
Likewise, the same procedure is done to find the second special determinant A2,
			2 22
		A2 = 
			-1 53
		(A2( = 2 (53) - (-1) (22) = 128
	We have now determined:
					(A( = 16
					(A1( = -208
					(A2( = 128
 Using:
		xi = (Ai(	
			(A(
	
	 we get, 
			 (A1(	 -208
			x1 = ( A( = 16 = -13 (Solution)
				(A2(	 128	
			x2 = (A(	 = 16 = 8 	 (Solution)	
Applying Cramer’s Rule in a 3x3 example
Using Cramer’s Rule to solve for the unknowns in three linear equations:
		5x​1​ - 2x​2​ + 3x​3​ = 16
		2x​1​ + 3x​2​ - 5x​3 ​​= 2
		4x​1​ - 5x​2​ + 6x​3​ = 7
	Then, 
		5 -2 3 x​1 ​16
		2 3 -5 x​2 = 2	
		4 -5 6 x​3​​ 7
				 5	-2 3 
The primary determinant (A(= 2 3 -5 = 5(18 - 25) + 2(12 + 20) + 3(-10 - 12) = - 37 
				 4 -5 6 	 
The three special determinants are:
			 16 -2 3	 
		(A​1​(= 2 3 -5 = 16(18 - 25) + 2(12 + 35) + 3(-10 - 21) = -111 	
			 7 -5 6	
			 5 16 3 
(A​2​​(= 2 2 -5 = 5(12 + 35) - 16(12 + 20) + 3(14 - 8) = -259
			 4 7 6
			 5 -2 16
		(A​3(= 2 3 2 = 5(21 + 10) + 2(14 - 8) + 16(-10 - 12) = -185
			 4 -5 7
Applying Cramer’s Rule:
		 
 ( A​1​(	 -111
		x​1​ = A = -37 	= 3 
		 (A​2​​( -259
		x​2​ ​= A = -37	= 7
		 (A​3​( -185
		x​3​ = A	= -37	 = 5
Applying Cramer’s Rule to Obtain Comparative Static Multipliers for an IS-LM System
A system of equations can always be presented in the following form,
 
�
where the n equations implicitly define a set of n functions which determine each of the n endogenous variables (y1,...,yn) in terms of the m exogenous variables (x1,...,xm). 
	For example, consider the IS equation 
and LM equation 
. Assume the output level 
and real interest rate level 
are endogenous.Assume government policy variables are exogenous (because they are controlled by government), including the level of government purchases 
 , the level of taxes 
, and the money supply level 
. Assume the level of net exports 
is exogenous, along with the expectations parameter 
 and the price level 
. We can rewrite these IS and LM equations as the system
 
		
	If some technical details are satisfied�, the implicit function theorem tells us that the endogenous variables 
and 
are determined implicitly as functions of the exogenous variables 
, if the Jacobian determinant is not equal to zero. For the general system, the Jacobian determinant is given by 
	
�
For our IS-LM example, the Jacobian determinant is 
.
Under the assumptions 
 (money demand increases when output increases), 
 (money demand decreases when the real interest rate increases) , 
 (the marginal propensity to consume is positive, but people also save), this determinant is negative. The non-zero determinant indicates that our system has a solution, and we will be able to use this determinant when we apply Cramer’s rule below to find comparative static multipliers. 
	Because the consumption function, investment function, and money demand function are all general functional forms, we have no hope of actually solving our IS-LM system. However, we can nonetheless learn how the exogenous variables impact the endogenous variables. When an exogenous variable changes, the equilibrium described by the IS and LM equations is perturbed. However, the assumption that the IS and LM equations always hold implies that the endogenous variables Y and r must adjust to new equilibrium levels. Comparative static analysis involves comparing equilibrium states before and after the change of an exogenous variable. The analysis is static because time plays no essential role. That is, we do not consider the disequilibrium path the economy might follow as it moves from one equilibrium to another. This is unfortunate since many interesting things happen can happen out of equilibrium. However, ignoring disequilibrium greatly simplifies the analysis while still providing a forecast of change consistent with the assumption that the new equilibrium will be reached. 
 	To obtain comparative statics multipliers for our system, the next step is to totally differentiate the system. Doing so, one obtains
	
�
Notice that differentiating a system in “the levels” yields a new system in “the differentials.” Our new system contains two equations and two endogenous differentials
---dY and dr. This system is linear because the differentials dY and dr only appear as coefficients. (There is no 
, for example, and no 
 as another nonlinear example.) There are multiple ways in to solve this linear system. One method is by using substitution. Using substitution is easier for systems with fewer equations and fewer variables. However, it can become a tedious art for more complicated systems. Cramer’s Rule is a general, systematic method for solving a linear system.
	To use Cramer’s rule, we must rearrange the linear system so it is the matrix form Ax=d. The matrix A has n rows and n columns, corresponding to the system’s n equations and n endogenous variables. The vector x has n rows and consists of the system’s n endogenous variables. The vector d has n rows and consists of all terms which do not contain an endogenous variable. Our system written in this matrix form is
�
The next step in using Cramer’s rule is to calculate the determinant 
�. Notice this determinant is the Jacobian that we calculated above. From above, we know that 
.
	The next step involves calculating the determinants 
 and 
, where the matrix Ai is created by replacing column i in the matrix A by the vector d. For our system the determinants 
 and 
 are given by
	
 
�
and
	
 
�
 The final step in using Cramer’s rule is to construct the solutions for the endogenous variables. In general, and for our example system, this is done as follows:
	
 
�
 and
	
 
�
	The comparative static multipliers are the coefficients on the exogenous differentials. Because 
�, the sign of the particular multiplier is the opposite sign of the coefficient’s numerator. There are a total of 12 multipliers for our system, showing how each of the six exogenous variables impacts each of the two endogenous variables at the margin. To illustrate what we can learn, assume that the only exogenous variable that changes is the government purchases variable
. This implies 
, 
, 
, 
, and 
 are all equal to zero. The solution equations above then become 
 
and 
.
This implies 
	
	and 
.
These results are actually partial derivative results because 
, 
, 
, 
, and 
 are all equal to zero. So, they are more appropriately written as such: 
	and 	
	The restrictions 
, 
, and 
 imply the two multipliers are positive as shown. These results are telling us that an increase in government purchases will tend to increase the level of output and increase the real interest rate level. The effect on output is the standard Keynesian “multiplier effect,” while the effect on the real interest rate is the standard “crowding out” effect. The extent to which there is a multiplier effect versus a crowding out effect depends upon the magnitudes of 
, 
, and 
. A “liquidity trap” occurs as 
 becomes large in magnitude (i.e., very negative), and the multiplier 
 approaches the simple Keynesian spending multiplier 
. This is because the multiplier 
 approaches zero, so no increase in the interest rate level occurs to crowd out investment spending. Alternatively, as 
 approaches zero, so money demand is not affected by the interest rate level, which was the Classical assumption prior to Keynes, the multiplier 
 approaches zero. This is because the multiplier 
 approaches infinity, so that the slightest increase in government purchases increases the interest rate level so much that the increase in government purchases is exactly offset by a decrease in investment. 
� See Fundamental Methods of Mathematical Economics by Alpha Chiang for a careful, understandable presentation of the implicit function theorem.
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