An engineer checks matrix properties while scaling a stiffness matrix. Given a11 (a11) = 2.5000; a22 (a22) = 3.0000; scalar (k) = 1.2000, determine the trace of kA (trkA).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except trkA is given, so isolate trkA symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for trkA:

   $$trkA = k (a_{11}+a_{22})$$

3. Step 3

   $$List the givens: a11 (a11) = 2.5000, a22 (a22) = 3.0000, scalar (k) = 1.2000$$

4. Step 4 — Substitute the given values:

   $$trkA = 1.2000 (2.5000+3.0000)$$

5. Step 5 — Evaluate:

   $$trkA = 6.6000$$

6. Step 6 — Check: returning trkA = 6.6000 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 13.2000 — kept a factor of two that cancels in the correct rearrangement.
• 3.3000 — dropped that same factor in the other direction.
• 7.2600 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

A student verifies the matrix property tr(kA) = k tr(A) numerically. Given a11 (a11) = 2.0000; a22 (a22) = 7.0000; trace of kA (trkA) = 98.0000, determine the scalar (k).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \dfrac{\text{tr}(kA)}{a_{11}+a_{22}}$$

3. Step 3

   $$List the givens: a11 (a11) = 2.0000, a22 (a22) = 7.0000, trace of kA (trkA) = 98.0000$$

4. Step 4 — Substitute the given values:

   $$k = \dfrac{\text{tr}(kA)}{2.0000+7.0000}$$

5. Step 5 — Evaluate:

   $$k = 10.8889$$

6. Step 6 — Check: returning k = 10.8889 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 21.7778 — kept a factor of two that cancels in the correct rearrangement.
• 5.4444 — dropped that same factor in the other direction.
• 11.9778 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

The matrix properties of a scaled transformation matrix are confirmed. Given a22 (a22) = 6.0000; scalar (k) = 4.9000; trace of kA (trkA) = 94.8000, determine the a11 (a11).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except a11 is given, so isolate a11 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for a11:

   $$a_{11} = \dfrac{\text{tr}(kA)}{k} - a_{22}$$

3. Step 3

   $$List the givens: a22 (a22) = 6.0000, scalar (k) = 4.9000, trace of kA (trkA) = 94.8000$$

4. Step 4 — Substitute the given values:

   $$a_{11} = \dfrac{\text{tr}(kA)}{4.9000} - 6.0000$$

5. Step 5 — Evaluate:

   $$a_{11} = 13.3469$$

6. Step 6 — Check: returning a11 = 13.3469 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 26.6939 — kept a factor of two that cancels in the correct rearrangement.
• 6.6735 — dropped that same factor in the other direction.
• 14.6816 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

An engineer checks matrix properties while scaling a stiffness matrix. Given a11 (a11) = 5.5000; a22 (a22) = 5.5000; scalar (k) = 0.8000, determine the trace of kA (trkA).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except trkA is given, so isolate trkA symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for trkA:

   $$trkA = k (a_{11}+a_{22})$$

3. Step 3

   $$List the givens: a11 (a11) = 5.5000, a22 (a22) = 5.5000, scalar (k) = 0.8000$$

4. Step 4 — Substitute the given values:

   $$trkA = 0.8000 (5.5000+5.5000)$$

5. Step 5 — Evaluate:

   $$trkA = 8.8000$$

6. Step 6 — Check: returning trkA = 8.8000 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 17.6000 — kept a factor of two that cancels in the correct rearrangement.
• 4.4000 — dropped that same factor in the other direction.
• 9.6800 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

A student verifies the matrix property tr(kA) = k tr(A) numerically. Given a11 (a11) = 7.5000; a22 (a22) = 9.5000; trace of kA (trkA) = 41.7000, determine the scalar (k).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \dfrac{\text{tr}(kA)}{a_{11}+a_{22}}$$

3. Step 3

   $$List the givens: a11 (a11) = 7.5000, a22 (a22) = 9.5000, trace of kA (trkA) = 41.7000$$

4. Step 4 — Substitute the given values:

   $$k = \dfrac{\text{tr}(kA)}{7.5000+9.5000}$$

5. Step 5 — Evaluate:

   $$k = 2.4529$$

6. Step 6 — Check: returning k = 2.4529 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 4.9059 — kept a factor of two that cancels in the correct rearrangement.
• 1.2265 — dropped that same factor in the other direction.
• 2.6982 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

The matrix properties of a scaled transformation matrix are confirmed. Given a22 (a22) = 4.5000; scalar (k) = 1.5000; trace of kA (trkA) = 46.9000, determine the a11 (a11).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except a11 is given, so isolate a11 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for a11:

   $$a_{11} = \dfrac{\text{tr}(kA)}{k} - a_{22}$$

3. Step 3

   $$List the givens: a22 (a22) = 4.5000, scalar (k) = 1.5000, trace of kA (trkA) = 46.9000$$

4. Step 4 — Substitute the given values:

   $$a_{11} = \dfrac{\text{tr}(kA)}{1.5000} - 4.5000$$

5. Step 5 — Evaluate:

   $$a_{11} = 26.7667$$

6. Step 6 — Check: returning a11 = 26.7667 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 53.5333 — kept a factor of two that cancels in the correct rearrangement.
• 13.3833 — dropped that same factor in the other direction.
• 29.4433 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

An engineer checks matrix properties while scaling a stiffness matrix. Given a11 (a11) = 5.0000; a22 (a22) = 9.5000; scalar (k) = 1.9000, determine the trace of kA (trkA).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except trkA is given, so isolate trkA symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for trkA:

   $$trkA = k (a_{11}+a_{22})$$

3. Step 3

   $$List the givens: a11 (a11) = 5.0000, a22 (a22) = 9.5000, scalar (k) = 1.9000$$

4. Step 4 — Substitute the given values:

   $$trkA = 1.9000 (5.0000+9.5000)$$

5. Step 5 — Evaluate:

   $$trkA = 27.5500$$

6. Step 6 — Check: returning trkA = 27.5500 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 55.1000 — kept a factor of two that cancels in the correct rearrangement.
• 13.7750 — dropped that same factor in the other direction.
• 30.3050 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

A student verifies the matrix property tr(kA) = k tr(A) numerically. Given a11 (a11) = 6.5000; a22 (a22) = 7.0000; trace of kA (trkA) = 14.4000, determine the scalar (k).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \dfrac{\text{tr}(kA)}{a_{11}+a_{22}}$$

3. Step 3

   $$List the givens: a11 (a11) = 6.5000, a22 (a22) = 7.0000, trace of kA (trkA) = 14.4000$$

4. Step 4 — Substitute the given values:

   $$k = \dfrac{\text{tr}(kA)}{6.5000+7.0000}$$

5. Step 5 — Evaluate:

   $$k = 1.0667$$

6. Step 6 — Check: returning k = 1.0667 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.1333 — kept a factor of two that cancels in the correct rearrangement.
• 0.5333 — dropped that same factor in the other direction.
• 1.1733 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

The matrix properties of a scaled transformation matrix are confirmed. Given a22 (a22) = 4.5000; scalar (k) = 3.6000; trace of kA (trkA) = 19.9000, determine the a11 (a11).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except a11 is given, so isolate a11 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for a11:

   $$a_{11} = \dfrac{\text{tr}(kA)}{k} - a_{22}$$

3. Step 3

   $$List the givens: a22 (a22) = 4.5000, scalar (k) = 3.6000, trace of kA (trkA) = 19.9000$$

4. Step 4 — Substitute the given values:

   $$a_{11} = \dfrac{\text{tr}(kA)}{3.6000} - 4.5000$$

5. Step 5 — Evaluate:

   $$a_{11} = 1.0278$$

6. Step 6 — Check: returning a11 = 1.0278 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.0556 — kept a factor of two that cancels in the correct rearrangement.
• 0.5139 — dropped that same factor in the other direction.
• 1.1306 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties

An engineer checks matrix properties while scaling a stiffness matrix. Given a11 (a11) = 5.0000; a22 (a22) = 10.0000; scalar (k) = 2.0000, determine the trace of kA (trkA).

Start with the thinking

• The governing relation printed in this handbook section is Matrix properties — trace and scalar multiplication.
• Everything except trkA is given, so isolate trkA symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Matrix properties state that scaling a matrix by k scales its trace by the same factor k.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

2. Step 2 — Rearrange symbolically for trkA:

   $$trkA = k (a_{11}+a_{22})$$

3. Step 3

   $$List the givens: a11 (a11) = 5.0000, a22 (a22) = 10.0000, scalar (k) = 2.0000$$

4. Step 4 — Substitute the given values:

   $$trkA = 2.0000 (5.0000+10.0000)$$

5. Step 5 — Evaluate:

   $$trkA = 30.0000$$

6. Step 6 — Check: returning trkA = 30.0000 to

   $$\text{tr}(kA) = k \, \text{tr}(A)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 60.0000 — kept a factor of two that cancels in the correct rearrangement.
• 15.0000 — dropped that same factor in the other direction.
• 33.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Matrix Properties