How to Solve Matrix Equations

How to Solve Matrix Equations

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What is a Matrix Equation?

A matrix equation has the form AX = BAX = B, where AA is the coefficient matrix, X is the column matrix of variables, and BB is the column matrix of constants on the right side of the equations in a system.

Solving Matrix Equation?

Let's figure out what XX is in the equation AX = BAX = B. We did this by multiplying each side of the equation by (A−1A−1). (the inverse of AA)

AX = B ⇒ A−1 (AX) = A−1 BAX = B ⇒ A−1 (AX) = A−1 B

We know that A−1AA−1A equals II, where II is the same-order identity matrix as AA. So,

IX = A−1 BIX = A−1 B

Also we know that: IX = XIX = X. So,

X = A−1 BX = A−1 B

The answer to the matrix equation is found here. This is also called the "inverse matrix equation," and the way to solve a set of equations by using the above formula is called the "inverse matrix method." So, here are the steps to use matrices to solve a set of equations:

  • Put the system into the form of the matrix equation AX = BAX = B.
  • Find the Inverse of AA.
  • To get the answer, multiply it by the constant matrix BB. i.e., X = A−1BX = A−1B.

Free printable Worksheets

Exercises for Solving Matrix Equations

1) Find the answer: X + [2−5 73] = [−4−2 4−9]

2) Find the answer: X − [113 −58] = [1321 −76]

3) Find the answer: 2Y + [9−23 −15−7] = [−8−27 133]

4) Find the answer: 2Y − [−1521 19−17] = [916 −127]

5) Find the answer: 2X + [−711 −7−7] = [15−29 17−11]

6) Find the answer: 2X + [19−208] = [25−31−12]

7) Find the answer: X×[2−4 13] = [3−1 2−3]

8) Find the answer: X×[51 2−3] = [86 912]

9) Find the answer: −3X = [−1833−69]

10) Find the answer: 7 Y = [7042−56]

 

1) Find the answer: X + [2−5 73] = [−4−2 4−9]

X = [−4−2 4−9] − [2−5 73] = [−63 −3−12]

2) Find the answer: X − [113 −58] = [1321 −76]

X = [113 −58] + [1321 −76] = [2424 −1214]

3) Find the answer: 2Y + [9−23 −15−7] = [−8−27 133]

2Y = [−8−27 133] − [9−23 −15−7] = [−17−4 2810] ⇒ Y = [−8.5−2 145]

4) Find the answer: 2Y − [−1521 19−17] = [916 −127]

2Y = [−1521 19−17] + [916 −127] = [−637 7−10] ⇒ Y = [−318.5 3.5−5]

5) Find the answer: 2X + [−711 −7−7] = [15−29 17−11]

2X = [15−29 17−11] − [−711 −7−7] = [22−40 24−4] ⇒ X = [11−20 12−2]

6) Find the answer: 2X + [19−208] = [25−31−12]

2X = [25−31−12] − [19−208] = [6−11−20] ⇒ X = [3−5.5−10]

7) Find the answer: X×[2−4 13] = [3−1 2−3]

X×[2−4 13] = [3−1 2−3] ⇒ X = [3−1 2−3]×[2−4 13]−1  = [3−1 2−3]×[31025 −11015]= [1710−32 110−12]

8) Find the answer: X×[51 2−3] = [86 912]

X×[51 2−3] = [86 912] ⇒ X = [86 912]×[51 2−3]−1  = [86 912]×[317117 217−517]= [33173017 −2917−4817]

9) Find the answer: −3X = [−1833−69]

−3X = [−1833−69] ⇒ X = [6−1123]

10) Find the answer: 7 Y = [7042−56]

7 Y = [7042−56] ⇒ Y = [106−8]

Matrix Equations Practice Quiz